Rail services have also been affected by the weather, with train speeds limited in many areas on Friday due to heavy rain overnight.
A Met Office amber alert for heavy rain in parts of Dumfries and Galloway as well as the Scottish Borders is in force until midday, while yellow weather warnings of snow and ice cover other areas of the country.
Forecasters said around 40-50mm of rain is expected in the amber warning area, with 60-70mm possible in upland locations, and they warned fast-flowing or deep floodwater is likely, causing danger to life.
The Scottish Environmental Protection Agency has issued 10 flood alerts and 27 flood warnings across Scotland.
River levels were expected to rise as a result of heavy overnight rain, with levels peaking during daylight hours on Friday.
Dumfries and Galloway Council urged people to take care, and tweeted: "We've seen a lot of rain overnight. There's a lot of surface water on roads across D&G. Please be safe out there if you're travelling."
Roads affected by flooding on Friday morning included the A90 northbound Stonehaven-Muchalls in Aberdeenshire, A90 southbound at Stracathro in Angus, with flood units responding in the area, and the A76 Sanquhar carriageway in Dumfries and Galloway, where Traffic Scotland said conditions are "extremely difficult".
Police in Dumfries said they were expecting flooding in Whitesands, Dockhead, and Dock Park overnight and on Friday.
Network Rail said the heavy overnight rain "means we need to limit train speeds in many areas of Scotland for everyone's safety".
ScotRail advised that services through Drumchapel station in Glasgow are being severely affected by the weather, and various other routes have also been affected, including trains going through Glasgow Queen Street.
Some trains may travel at a reduced speed or be cancelled.
The Met Office said the deadly bomb cyclone that sent temperatures plunging in the US over Christmas is causing the unsettled weather in the UK.
A yellow weather warning for heavy rain is in effect in Scotland until 2pm on Friday, covering central Scotland, Tayside and Fife, south-west Scotland, Lothian and Borders and Strathclyde.
A yellow warning of rain for Northern Ireland expired at 10am on Friday.
A yellow warning of snow and ice covers northern Scotland, apart from Orkney and Shetland, until 9pm on Friday.
Police Scotland said there is a high risk of disruption and urged people to plan ahead and drive to the conditions if they have to travel.
The Scottish Government said the Multi-Agency Response Team will monitor conditions throughout the amber warning period.
Superintendent Stewart Mackie, of Police Scotland's Road Policing Division, said: "An amber weather warning has been issued for heavy rainfall and flooding across southern Scotland, and people should consider delaying any travel plans until conditions improve.
"If your journey is essential, please plan ahead by making sure you have sufficient fuel and supplies in your vehicle, a charged mobile telephone and always drive as the conditions allow.
"Be aware road closures and travel delays are likely, with potential disruption to public transport services."