The arrival of Storm Franklin marks the first time the nation has had three named storms in a week.
Winds of up to 75pmh have been forecast with the strongest winds hitting the Hebrides and Argyll, Ayrshire and Dumfries and Galloway. Gusts of 80mph are expected on the coasts.
Some 18 flood warnings and seven alerts have been issued across the Scottish Borders, Ayrshire, Orkney and the Western Isles by the Scottish Environment Protection Agency (Sepa).
Katharine Smith, flood duty manager at the Environment Agency, urged people to “stay away from swollen rivers” while teams deploy temporary barriers and pumps on the river.
“We advise people to stay away from swollen rivers and not to drive through flood water as just 30cm of flowing water is enough to move your car.”
Across the UK a number of flood warnings were also in place, including two rare “severe” warnings where rainfall could also pose a “danger to life” for communities along the River Mersey in Greater Manchester.
Met Office meteorologist Becky Mitchell said last week marked the first time three named storms have been recorded within seven days since the storm-naming system began in 2015, with Dudley, Eunice and Franklin.
She added that there will “definitely be some impact” from Storm Franklin on Monday but it is not expected to be “as severe” as Eunice because the strongest winds will be confined to the coast.
The River Don burst its banks in the Sprotbrough area of Doncaster in South Yorkshire on Sunday night, and police have warned people to stay away from dangerous “fast flowing” water.
Train operator Northern issued a “do not travel” alert to passengers due to the impact of storms.
There is severe disruption on many of its routes, including between Sheffield and Manchester, Scarborough and Hull, York and Sheffield, Bradford and Leeds, Chester and Manchester, and Carlisle and Newcastle.
Rail operator TransPennine Express told customers on Twitter: “Due to a river flooding the railway at Mexborough, trains are currently unable to operate between Doncaster, Meadowhall and Sheffield. All lines are blocked.
“This is preventing road transport in the area. Customers are advised only to travel if their journey is essential.”