Around 5,000 Scottish Hydro Electric Power Distribution (SHEPD) customers remained offline at lunchtime, down from 6,000 earlier in the morning as engineers worked to restore supplies.
The Western Isles, Fort William and Oban areas have been worst affected, while engineers are working to repair a number of smaller faults across the network area, mainly in the Highlands.
Strong winds swept across the country last night with gusts of 90mph on South Uist and 55mph in Glasgow.
The strongest wind speed was recorded on Cairnwell mountain in the Cairngorms, which was hit by a 120mph gust at 2am.
Three passengers were injured on the MV Hrossey ferry from Shetland to Aberdeen at around 11pm last night as it sailed through rough seas and heavy swell.
Two of them needed hospital treatment in Kirkwall, Orkney, for non life-threatening injuries. The ship left Orkney at around 8.10am this morning to continue its journey to Aberdeen.
A number of passenger vehicles were also damaged and their owners have been informed.
In the Western Isles, Tolsta School on Lewis was shut today due to storm damage, while Laxdale School on Lewis and Leverburgh School in Harris were closed due to lack of power.
The Met Office had issued amber ''be prepared'' warnings of high wind for the Highlands, Western Isles and Strathclyde, and less severe yellow ''be aware'' warnings for the rest of Scotland overnight into this morning.
Nine flood alerts and one flood warning were also in place today due to heavy rain.
SHEPD said: "Our engineers have been working since the early hours to repair damage to the electricity network in the north of Scotland following the high winds.
"A number of customers were restored overnight and good progress continues to be made with around 1,000 customers having their power restored this morning. Our engineers continue to work hard in difficult conditions to restore around 5,000 customers who remain without power.
"We anticipate restoring the majority of customers by this afternoon and only very difficult areas possibly remaining into this evening."
The Erskine and Forth Road bridges were closed to high-sided vehicles last night as winds picked up, while trains between Glasgow and Ardrossan were disrupted and a host of roadswere affected by surface water and debris blown down by the winds.
Forecasters predicted that the winds will lessen today but the weather is expected to remain wet and windy in the run-up to Christmas.
John Griffiths, a forecaster with MeteoGroup, said: "The winds have eased and will continue to ease throughout the day but it's still very windy."
The latest storms came just two weeks after the last serious weather front saw a lorry driver killed when his vehicle was blown over on to two cars.
The rail network also ground to a halt as trees, hay stacks and even trampolines were blown on to train tracks across the country.