Gusts of more than 80mph have hit exposed parts, with the west and south-west of Scotland worst affected.
Properties in Dumfries and Galloway and the north have lost electricity as trees and debris brought down power lines in the regions.
Thirteen flood warnings are in place in Tayside with as much as 40mm of rain expected in some places.
Rainfall on already-saturated ground is likely to flood low-lying agricultural areas but the risk of community flooding is low, according to the Scottish Environment Protection Agency.
The strongest gusts of wind have been in Inverbervie, Aberdeenshire (81mph) and in Dundrennan, Dumfries and Galloway (70mph).
On the mountain tops the wind reached 93mph in the Cairngorms at 10am today and 75mph in Glen Ogle near Stirling.
Scottish Power said 2,000 of its customers remain cut off in Dumfries and Galloway, down from 6,500 earlier in the day.
''The high winds have brought down trees on to overhead lines cutting power supplies,'' a spokesman for the company said.
''The weather conditions and road closures are making it difficult for engineers to get to the affected areas but we will do all we can to get people back on the grid as soon as we can.''
Scottish and Southern Energy said 400 homes are still without electricity in northern parts of the country, down from 1,200.
Just over half of these are in the Orkney Islands, with properties in Angus, Aberdeenshire and Inverness also without power.
Engineers are expected to reconnect all those affected by this evening.
Many roads have been hit by flooding and fallen trees and the Forth and Tay road bridges are closed to high-sided vehicles. Police are urging people across the country to take extra care if driving.
Rail services have been disrupted, with Scotrail warning trains to and from Aberdeen may be delayed or revised at short notice.
Services between Glasgow Central and Ardrossan Harbour and Glasgow Central and Largs are also affected by cancellations and delays.
Some CalMac ferry services have also been cancelled.
John Griffiths, a forecaster for MeteoGroup UK, said the high winds would begin to ease over the next few hours.
"At the moment there are still strong winds in southern parts with the eye of the storm just above Scotland," he said.
''The low pressure system is starting to move away which is why later tonight the winds will ease off as it heads off towards Norway.
"Tomorrow is going to be a little bit drier and brighter although there will still be showers in western parts."
Transport Minister Keith Brown said: ''In Scotland, the south-west is being worst affected by today's heavy rain and strong winds, although conditions are not as severe as the storms that affected the north and north-west earlier this week.
''And these windy and wet conditions are not unusual for this time of year. However, we must not be complacent, which is why the Scottish Government's resilience committee is closely monitoring the situation, working closely with all our partners to share information and ensure resources are targeted where they are needed.
''The risk of flooding appears to be subsiding, although the stormy conditions continue to disrupt the rail network and power supplies. Engineers are working to resolve these issues as soon as possible.''
On Christmas Eve, a storm caused widespread disruption that stretched into Christmas Day in some areas.
The travel plans of hundreds were affected and as many as 3,500 homes lost their electricity supply.