The UK has been hit by the fringes of the storm, with heavy rain and winds set to persist throughout the weekend.
In France, Nice and its surrounding areas were the hardest hit overnight as 500mm (20ins) of rain was recorded over 12 hours in some areas, according to Meteo France.
That amounts to nearly a year's worth of average rainfall, more than on 3 October 2015 when 20 people died in floods in Cannes and the surrounding area on the French Riviera.
In the village of Sambughetto, in the northwest Italian region of Piedmont, a record 630mm (24.8ins) of rain fell in 24 hours.
Roads have been swept away and homes have been badly damaged in the mountainous region on the border of France and Italy.
French authorities said firefighters were looking for at least eight people who are missing.
They include two firefighters whose vehicle was carried away by a swollen river.
Christian Estrosi, Nice's mayor, expressed his "sympathy" for the families of those missing and said the flooding was the worst in the area for more than a century, adding that he was shocked by what he saw when he flew over the scene in a helicopter.
He said more than 100 homes have been destroyed or severely damaged.
Firefighters said several dozen people were evacuated from their homes overnight.
French President Emmanuel Macron said he is thinking of the victims and their loved ones as he expressed his gratitude for rescue workers.
"Together, we will get through this," he tweeted.
French Prime Minister Jean Castex and Interior Minister Gerald Darmanin are expected to arrive in the area later on Saturday.
The storm caused heavy winds of more than 112mph (180kph) in Brittany between Thursday and Friday.
In Italy, a firefighter was killed after he was hit by a falling tree and another man died after his car was swept into a river when the road subsided. Another Italian was missing as night fell.
There were fears the storm would bring flooding, with the water level in the Po river, one of northern Italy's biggest rivers, rising by 3 metres.
In Venice, the Italian city's long-delayed flood barriers were deployed for the first time on Saturday.
Forecasters warned the storm could combine with high tides to inundate the sinking city.
The network of 78 bright yellow barriers that guard the entrance to the Venetian lagoon started to lift from the seabed more than three hours before high tide was scheduled.
Driven by strong winds and heavy rain, the tide was expected to reach 130cm (51ins), well below the 187cm (74ins) tide that plunged much of Venice under water last November, but enough to leave low-lying areas deep under water.
The barrier system, called Mose, was designed in 1984 and was due to come into service in 2011 but the project was plagued by corruption, overrunning costs and prolonged delays.
It successfully protected the city from major flooding, bringing relief after years of inundations.