Two people had to be rescued after their car was swept off the road near Newcastleton, in the Scottish Borders.
In Ilkley and Calder in West Yorkshire the Army has been helping residents shore up flood defences, a week after they were hit by Storm Ciara.
The Environment Agency has warned flooding could be worse than Ciara because of the saturated ground.
Last weekend's storm caused hundreds of homes to be flooded and left more than 500,000 people without power.
As of Sunday morning, three severe flood warnings were in place across Scotland for the Hawick area in the Scottish Borders - with some people advised to evacuate their homes.
There is also one severe flood warning in Wales.
Meanwhile, more than 100 flood warnings are in place across England, more than 40 in Scotland and more than 60 in Wales.
In York, the Environment Agency has predicted the River Ouse could come close to record levels, last seen in 2000.
Some 75 soldiers from 4th Battalion, Royal Regiment of Scotland, have been helping build flood barriers and repair defences in Ilkley and Calder in West Yorkshire.
A further 70 Reservists from 4th Battalion, The Yorkshire Regiment, will also be providing support where required.
Separate flood warnings and advice have also been issued for residents in the Borders, Dumfries and Galloway, South Lanarkshire and South Ayrshire.
Across the UK road, rail and air travellers also face disruption, with British Airways and easyJet flights among those affected.
Amber warnings for rain and yellow warnings for wind are in place for most of the country into Sunday evening.
This means flooding could cause a danger to life, power cuts are expected and there is a good chance transport links will be impacted.
On Saturday, the body of a man was pulled from the sea off the Kent coast.
The man was declared dead at the scene in Herne Bay after emergency services were called at 12:15 GMT. Kent Police are not linking his death to Storm Dennis.
The force said it was not known how the man had entered the water and his death was "not being treated as suspicious".
A second body was found by the RNLI at about 13:00 GMT on Saturday after a seven-hour search in "rough seas" for a man who fell from a fuel tanker off the coast of Margate.
In other developments:
EasyJet has cancelled about 350 flights over the weekend - almost 100 of these are to and from London's Gatwick Airport
About 60 flights were grounded at London's Heathrow Airport. Most of them are British Airways
Rail passengers across the country were urged to check before travelling, with delays and cancellations expected on certain routes
For more information, check the BBC Weather website and your BBC Local Radio station for regular updates
The Met Office has forecast that some areas could see between 120-140mm of rainfall and gusts of up to 80mph over the weekend.
The predictions are not as severe as last weekend when Ciara brought as much as 184mm of rain and gusts reaching 97mph.
But experts have warned Storm Dennis could cause more flooding damage, as already saturated ground is met with a "perfect storm" of heavy rain, strong winds and melting snow.