Britain, Italy and Spain were also sweltering in extreme heat on Tuesday, with record temperatures in some regions disrupting schools and transport networks.
Europe is warming at more than twice the global average, according to the World Meteorological Organisation, making such prolonged heat episodes increasingly likely.
Much of France is under severe heat alert and set to experience temperatures about 40C on Tuesday, Meteo France said, with temperatures of up to 43C expected in some parts of western France.
The country has just recorded its hottest afternoon and night since records began in 1947.
Fifty-four departments are under red alert in what forecasters described as unprecedented.
Across France, people have been jumping into canals and rivers to cool off.
French sports minister Marina Ferrari said she understood the urge to escape the heat but warned against swimming in unauthorised or dangerous areas.
Speaking before an emergency meeting on the heatwave, French Prime Minister Sebastien Lecornu said: "A sad scourge when it comes to drownings, as the latest figures just reported to us show 40 deaths since June 18, most of them young people."
On Monday, first responders were unable to resuscitate two children, aged two and four, who were found unconscious by their mother in the family car outside their home, said a prosecutor in Carpentras, southeast France.
In Paris, commuters struggled through sweltering conditions after sleepless nights in apartments ill-equipped for heat.
Some trains were cancelled, including between Paris and Brussels.
The heatwave is driven by a weather pattern known as an Omega block, with a bulge of hot air in the middle and cooler air either side, allowing temperatures to build day after day.
Heatwaves and storms are being intensified by climate change, pushing temperatures higher and causing more rainfall.
In Italy, the health ministry issued its highest level alert for 15 cities and authorities took measures to curtail work.
Storms are expected later on Tuesday over the Alps and Apennines, bringing heavy rain, gusty winds and hail.
Britain is also in the grip of the heat, with the Met Office forecasting temperatures of up to 37C in southern England on Tuesday - potentially a new June record - before rising further on Wednesday and Thursday.
Dozens of schools have said they will close early, due to old buildings not being suitable for classrooms with more than 30 children in.
Spain's meteorological agency has issued red alerts across parts of the country, warning of dangerous heat with temperatures expected to reach 44C.
The warnings follow an already extreme day on Monday, including a peak above 45C in Andujar.
Night-time has brought little relief, with temperatures above 25C recorded early on Tuesday.
In Belgium, soaring temperatures forced a primary school in Tervuren, near Brussels, to relocate its final exams to a nearby church.
Transport networks across Europe came under strain.
Britain's Network Rail warned passengers to travel only if necessary later this week as temperatures approach 39C, with speed restrictions likely to disrupt services.
In London, overnight thunderstorms - part of the same volatile weather pattern - caused further disruption, including at Heathrow Airport.
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