BERLIN — Days before roiling waters tore through western Germany, a European weather agency issued an “extreme” flood warning after detailed models showed storms that threatened to send rivers surging to levels that a German meteorologist said on Friday had not been seen in 500 or even 1,000 years.

By Friday those predictions proved devastatingly accurate, with at least 125 people dead and 1,300 unaccounted for, as helicopter rescue crews plucked marooned residents from villages inundated sometimes within minutes, raising questions about lapses in Germany’s elaborate flood warning system.

Numerous areas, victims and officials said, were caught unprepared when normally placid brooks and streams turned into torrents that swept away cars, houses and bridges and everything else in their paths.

“It went so fast. You tried to do something, and it was already too late,” a resident of Schuld told Germany’s ARD public television, after the Ahr River swelled its banks, ripping apart tidy wood-framed houses and sending vehicles bobbing like bath toys.

Extreme downpours like the ones that occurred in Germany are one of the most visible signs that the climate is changing as a result of warming caused by greenhouse gas emissions. Studies have shown a warmer atmosphere can hold more moisture, generating more, and more powerful, rainfall.

The floods that cut a wide path of destruction this week through Germany, Belgium, Switzerland and the Netherlands were bigger than any on record, according to meteorologists and German officials.

German officials said Friday their warning system, which includes a network of sensors that measure river levels in real time, functioned as it was supposed to. The problem, they said, was an amount of rain they had never seen before — falling so rapidly that it engorged even small streams and rivers not normally considered threats.

To describe the events of recent days as a 100-year flood would be an understatement, said Uwe Kirsche, a spokesman for the German Weather Service.

“With these small rivers, they have never experienced anything like that,” Mr. Kirsche said. “Nobody could prepare because no one expected something like this.”

On Tuesday, Felix Dietsch, a meteorologist for the German Weather Service, went on YouTube to warn that some areas of southwest Germany could receive previously unimaginable volumes of rain.

The weather service, a government agency, assigned its most extreme storm warning, code purple, to the Eifel and Mosel regions, one of numerous government warnings issued on Twitter and other media earlier this week and transmitted to state and local officials.

But the waters rose so swiftly that some communities’ response plans were insufficient while others were caught off guard entirely.

Medard Roth, the mayor of Kordel, in the hard-hit state of Rhineland-Palatinate, said that he activated his town’s emergency flood response once Kyll River approached dangerous water levels. But the waters rose too rapidly to be held back by the usual measures.

“By 6 p.m., everything was already under water,” Mr. Roth told Bild, a German newspaper. “Nobody could have predicted that.”

Ursula Heinen-Esser, the environment minister for the state of North Rhine-Westphalia, said on Friday that floodwaters had reached “levels never before recorded.”

The German flood warning system leaves it up to local officials to decide what action to take, on the theory that they are best informed about local terrain and what people or property lies in the path of an overflowing river.

In the Ahrweiler district of neighboring Rhineland-Palatinate, regional officials issued their first warning to residents living near the banks of the river as it approached its record level of 3 meters, or nearly 10 feet. Three hours later, a state of emergency was declared.

By that time, many people had fled to the upper levels of their homes, and those who could not move fast enough died, including 12 handicapped people in an assisted living home in Sinzig.

“The warnings arrived,” Mr. Kirsche of the German Weather Service said. “But the question is why didn’t evacuations take place sooner? That’s something we have to think about.”

MÜSCH, Germany — The bridge that spans the River Ahr washed away last night at around 10:00, said Michael Stoffels, 32, whose own house got flooded by about 12 feet of water.

Müsch, a village of 220 people at the junction of the Ahr and Trierbach rivers, was clobbered by the flash floods that have inundated this part of Germany. Only one person has died, but Müsch on Friday evening was without electricity, running water or cellphone coverage.

Residents and their friends were trying to clean up their battered homes, cracked streets and ruined cars. Local firefighters, like Nils Rademacher, 21, were managing the traffic of bulldozers, small trucks and backhoes, while instructing drivers that roads farther into the river valley were blocked with trees or made impassable by fallen bridges.

“A lot of good cars crashed or got crushed,’’ said Maria Vazquez, who works in a nearby auto repair shop. “I work with cars, so that’s sad, but I just hope that all the people are OK.”

The water rose to flood the village in less than two hours on Wednesday, and came halfway up the houses, Ms. Vazquez said.

The riverbanks were scenes of devastation, with crushed cars and thick tree stumps, while many of the cobbled streets were covered with mud and debris. Truckloads of broken furniture, tree branches and chunks of stone were being driven slowly over downed power lines.

The yellow road sign that tells drivers that they have entered Müsch was pulled out of the ground, laying bent and nearly adrift in the Trierbach River.

Mr. Stoffels said that he had no warning from the government, but that he rushed home from the retail store he manages nearby when a neighbor called. He was lucky, he said, since he has storage on the ground level and his living area is above that. The children’s playground next to his home, along the Ahr, was shattered, as was the main village electrical station, even before the bridge washed away.

He and his brother, who traveled 100 miles to help, and his friends, all wearing boots and muddy clothes, were trying to clean up as best they could. It helped, he said, that Müsch, in the Ahrweiler District of Rhineland-Palatinate close to the border with North Rhine-Westphalia, is farming country.

“Nearly everyone has a small tractor or a bulldozer of some kind,’’ he said. And it was true — the local firefighters were there, but there was little government presence, residents said. On Thursday, Mr. Stoffels said, “a couple of soldiers came for a time and a policeman looked around.”

Not far away, larger villages and towns were devastated, and more than 1,000 people are reported missing by the authorities.

Roger Lewentz, Rhineland-Palatinate’s interior minister, was unable to give an exact number of missing in his state.

“We do not yet know for sure whether some of them may be on vacation or simply unavailable. After all, the power and telephone connections are down in many affected locations,” he told Der Spiegel.

“There haven’t been floods like this here in 100 years,’’ said Sebastian Stich, 28, an office worker from nearby Barweiler who came to help his neighbors. “The bridges, the power, it’s all gone.’’

The floods devastating Europe have killed scores of people, leaving at least 1,300 missing, uprooting families, causing massive financial damage and reducing homes and cars to the state of floating bath toys. But it is not the first time the continent has been buffeted by a deluge. Here are some of the other major lethal floods and flooding caused by storms in recent years:

February and May 2014

A 7-year-old boy dead after falling ill in a flooded home in Surrey. A kayaker drowned on a swollen Welsh river. A coastal railroad ripped up by waves in Cornwall. In a matter of months in 2014, at least 5,000 houses in Britain were damaged in what was then seen as one of the rainiest seasons in nearly 250 years. While some blamed the flooding on the austerity measures of David Cameron, the prime minister at the time, others pointed to climate change. In May of that same year, the heaviest rains and floods in 120 years hit Bosnia and Serbia, killing at least 33 people, forcing thousands out of their homes, and cutting off power in 100,000 households in Serbia, as several months’ worth of rainfall fell in a matter of days.

June 2013

Germany is no stranger to flooding. In Bitterfeld, in eastern Germany, some 10,000 people were asked to leave their homes in June 2013 after a levee on the Mulde River burst, amid some of the worst flooding that some German regions had seen in centuries. More than 600 residents of Dresden were brought to safety as electricity and water services to the city’s affected center were cut off. Chancellor Angela Merkel, now tested by the current flooding, showed her mettle at the time, touring three of the hardest hit areas to wade through ankle-deep floodwaters and visit victims of the flood.

January 2007

The storm was called Kyrill by German meteorologists, and it spurred unrelenting rain in Britain, Ireland, France, Belgium and the Netherlands. The howling gale churned through the British Isles and Northern Europe, uprooting trees, shattering windows, flooding beaches and forcing the cancellation of hundreds of flights at airports from London to Frankfurt. According to the European Environment Agency, Kyrill killed 46 people and resulted in overall losses worth 8 billion euros. At the time, it was one of the most damaging extreme weather episodes ever recorded in Europe. The name Kyrill stemmed from a German practice of naming weather systems. Anyone may name one, for a fee, and three siblings had paid to name the system as a 65th birthday gift for their father, not realizing it would grow into a fierce storm.

August 2005

Such was the deluge in Central and Southern Europe in 2005 that in the Alps, military helicopters were deployed to ferry in supplies, evacuate stranded tourists and even stranded cows in mountain pastures threatened by rising water. The floods left dozens dead. In Romania, which was badly affected by the flooding, victims were drowned as torrents of water rushed into their homes. Austria, Bulgaria, Germany and Switzerland were also buffeted by the flooding. The scenes of devastation were visceral and shocking. The Aare River broke through the windows of a children’s clothes shop in Bern, leaving baby strollers and toys floating in muddy water. Much of the historic old city of Lucerne remained underwater. Meanwhile, in southern Poland, rivers broke their banks and at least seven bridges collapsed.

August 2002

In 2002, some of the worst rains since 1890 pelted the Czech Republic, putting part of the historic center of Prague underwater and resulting in 50,000 residents being ordered to evacuate, as rivers swelled by near constant rain. The death toll from the floods, which ravaged East and Central Europe, including Germany and Austria, and southern Russia, was more than 110. The flooding caused billions of dollars worth of damage. The floods helped propel Germany’s chancellor, Gerhard Schröder, to re-election because of his management of the crisis. In Austria, the Salzach River burst its banks south of Salzburg and threatened to inundate the city at the height of its famous summer festival, forcing the authorities to close most bridges and major roads. Floodwaters rose in Hungary and Germany, and in northern Austria the authorities halted river traffic on parts of the Danube.

Was the flooding caused by climate change?

Tying a single weather event to climate change requires extensive attribution analysis, and that takes time, but scientists know one thing for sure: Warmer air holds more moisture, and that makes it more likely that any given storm will produce more precipitation.

For every 1 Celsius degree of warming, in fact, air can hold 7 percent more moisture.

On average, the world has warmed by a little more than 1 degree Celsius (about 2 degrees Fahrenheit) since the 19th century, when societies began pumping huge amounts of greenhouse gases into the atmosphere.

“Any storm that comes along now has more moisture to work with,” said Jennifer Francis, a senior scientist with the Woodwell Climate Research Center in Massachusetts. “That’s the straightforward connection to the increased frequency of heavy downpours.”

And, although it is still a subject of debate, some scientists say climate change might be causing storms to linger longer.

Some studies suggest that rapid warming in the Arctic is affecting the jet stream. One consequence of that, said Hayley Fowler, a professor of climate change impacts at Newcastle University in England, is that the river of wind is weakening and slowing down at certain times during the year, including summer. And that, in turn, affects weather systems farther south.

“That means the storms have to move more slowly,” Dr. Fowler said. The storm that caused the flooding was practically stationary, she noted.

The combination of more moisture and a stalled storm system means a lot of rain can fall over a given area.

Geert Jan van Oldenborgh, one of the primary scientists with World Weather Attribution, a group that quickly analyzes specific extreme weather events to see whether they were made more likely, or not, by climate change, said the group was discussing whether they would study the German floods.

Beyond the speed of a weather system and its moisture content, there are many factors that affect flooding that can make an analysis difficult. Local topography has to be taken into account, as that can affect how much runoff gets into which rivers.

Human impacts can complicate the analysis even further. Development near rivers, for instance, can make runoff worse by reducing the amount of open land that can absorb rain. Infrastructure built to cope with heavy runoff and rising rivers may be under-designed and inadequate.

An increasingly hot, dry and deadly summer has gripped much of the Western United States, with heat claiming lives in the Pacific Northwest and Canada in record numbers, and a deepening drought threatening water supplies — all of which is setting the stage for another potentially catastrophic fire season in California and neighboring states.

A fourth major heat wave was forecast to roast parts of the region again this weekend. It comes two weeks after a record-shattering spate of high temperatures — which scientists said would been virtually impossible without climate change — killed hundreds of people in the United States and Canada.

A week ago, Death Valley hit a 130-degree high, matching a reading from last year that may be the highest reliably recorded temperature on earth. Also this past weekend, Las Vegas tied its record high, 117 degrees, and Grand Junction, Colo., topped its previous record, hitting 107 degrees.

At least 67 weather stations from Washington State through New Mexico have recorded their hottest temperatures ever this summer, the National Weather Service said this week. Those records stretched back at least 75 years.

The heat helped drive the rapid growth of a wildfire in southern Oregon, known as the Bootleg Fire, that has burned more than 240,000 acres — about a third the size of Rhode Island, America’s smallest state. The fire, the largest of dozens across the West, has destroyed about two dozen homes, threatens 1,900 more and has set off a wave of evacuations.

The fire also burned across a power line corridor that serves as a major contributor to the electrical grid in California, where officials have issued warnings this week asking residents to conserve power by turning up their thermostats and turning off appliances, or risk rolling blackouts.

One part of the West saw some relief from the crushing heat this week, as monsoon rains fell on the Southwest, including New Mexico and Arizona. But the result was yet another disaster: flash flooding that left some city streets in Arizona awash in muddy water and propelled a torrent of water through part of the Grand Canyon, washing away a camp where about 30 people on a rafting trip were spending the night, killing one.

As the Earth warms from climate change, heat waves are becoming hotter and more frequent. “And as bad as it might seem today,” Jonathan Overpeck, a climate scientist at the University of Michigan, recently told The New York Times, “this is about as good as it’s going to get if we don’t get global warming under control.”

A breach in the dike along the Juliana Canal in the southern Netherlands on Friday was closed by the Dutch military by dumping hundreds of sandbags into the growing hole. Hours before, thousands had been told to evacuate after the dike was breached along the canal, a 22-mile waterway that regulates the Meuse River.

The river’s water level is at heights not witnessed since 1911, the Dutch national broadcaster NOS reported.

That is no small thing is a water-logged country where taming water has been a matter of survival for centuries and the imperative to keep levels under control is inextricably bound up with Dutch identity. Much of the country sits below sea level and is gradually sinking. Climate change has also exacerbated the twin threats of storms and rising tides.

Residents of the villages of Brommelen, Bunde, Geulle and Voulwames were ordered to evacuate immediately, after initially being told to move to higher floors in their homes. About 10,000 people live in the area.

The local authorities said there was “a large hole” in the dike, prompting fears that the entire area would be flooded. While parts of the area were flooded, a disaster was averted after the breach was closed. NOS said the dike was still unstable and continued to be monitored.

Upriver, near the city of Venlo, evacuations were ordered for whole neighborhoods and surrounding villages, in total 10,700 people and 7,100 houses, the municipality said in a tweet. People have until 6 p.m. local time to leave their homes.

Record water levels are moving through the Meuse River, prompting evacuations and fresh inspections of dikes along the river that empties into the North Sea. The river is a key waterway for European shipping connections.

Following flooding in recent decades, the Dutch authorities have designated special areas that can be flooded with excess water when critical levels are reached.

The Netherlands has so far been spared much of the death and destruction that this week’s flooding has caused in Germany and Belgium. But in Valkenburg, a city in the south of the Netherlands with about 16,000 residents, damage was severe. Hundreds of houses were without power, and the center of the city was flooded.

“The damage is incalculable,” Mayor Daan Prevoo of Valkenburg told the Algemeen Dagblad newspaper. He predicted that repairs would take weeks.

In Liège, Belgium’s third-largest city, much of the early panic eased on Friday as residents said the waters of the Meuse river seemed to recede, at least a bit.

Fears that a major dam might break led the mayor to call for parts of the city to be evacuated late Thursday. But on Friday, people were allowed back, though they were told to stay away from the river, which was still lapping over its banks.

“The situation is now under control, and people can return to their homes,” Laurence Comminette, the spokeswoman for the mayor, said in an interview. “Of course not everyone can go back, because many homes have been destroyed. But there is no longer an imminent danger of more flooding.”

Georges Lousberg, 78, said he thought the crisis was largely over in the city. “It did not rain much today, and the weather is supposed to be better the rest of the week.”

He said there had been times when the Meuse was even higher, especially before walls were built along its banks. “The worst flooding was in 1926,” he said.

Prasanta Char, 34, a postdoctoral student in physics at the University of Liège, said he had been anxious about rain overnight after the mayor’s evacuation call.

He had gone looking to buy water, but had a hard time because so many stores were closed. He finally found a small convenience store in the shuttered city.

“It’s much worse in Germany, and a lot of the roads are shut and the trains are stopped,” he said, “I’m still a bit anxious about rain, but today it seems better.”

Forecasts predicting improved weather for Western Europe over the weekend offered some hope amid the deluge, potentially aiding search-and-rescue efforts in areas devastated by floods.

The heavy rain in Germany in the Ahrweiler district of Rhineland-Palatinate was forecast to let up later Friday and over the weekend, after flooding left 1,300 people unaccounted for in the region. Emergency workers put sandbags in place to stem the rising waters in the region’s remote villages, like Schuld, where heavy flows of water washed away six homes and left more close to collapse.

On Saturday and Sunday, there is about a 20 percent chance of rain in that area, and temperatures are expected to rise above 70 degrees Fahrenheit with partial sunshine later in the day, according to Weather.com. Conditions are likewise expected to improve in the state of North Rhine-Westphalia, also in western Germany, where at least 43 people have died in the flooding.

Andreas Friedrich, a meteorologist for Germany’s national weather service, said that dry, sunny weather was likely over the next few days in the western states hit by floods. The weather service has issued a warning about possible floods in the touristy area of southeastern Germany, north of the Alps, over the weekend, but conditions are not expected to be as bad as they were in the western part of the country, he said.

In Belgium, the weather is also expected to clear up over the weekend. The Royal Meteorological Institute of Belgium forecast only light rain in the hilly Ardennes region, which experienced heavy flooding over the past few days. In Liège, which was also hard hit, there was a 3 percent chance of precipitation on Saturday, according to the AccuWeather forecasting service.

Alex Dewalque, a spokesman for the meteorological institute, said water levels in the worst-hit parts of Belgium were already falling, making it easier for emergency workers to rescue stranded people and search for casualties. He said the coming days would be much drier and with warmer temperatures, and that there were no flood warnings.

More rain was expected in Switzerland’s northern Alps on Friday, however, and officials warned of more potential flooding in parts of the country. Lake Lucerne reached critical levels, forcing the closing of some bridges and roadways.

Sarah Schöpfer, a meteorologist at Switzerland’s Federal Office of Meteorology and Climatology MeteoSwiss, said she expected rainfall over the affected areas of Switzerland to lighten.

“We expect that tonight the precipitation activity weakens further and tomorrow it mainly affects the eastern Swiss Alps (mainly regions that did not get the highest amounts of rain during the last few days),” she said in an email. “So apart from the last showers today and tomorrow, the following days will be dry.”