The isle of Arran and parts of the south-west mainland of Scotland were particularly badly hit.
Power companies are working with councils to send generators to affected areas while engineers try to clear routes and restore energy.
A spokesman for Scottish Hydro said engineers are working to get people back on the grid today, but that access is not possible to all parts of the island.
"Mobile generators are on the island and more are en route but access is proving difficult," he said.
"We're trying everything we can to get people back on today but we can't give an indication of when everyone will have their power restored."
Whiting Bay Hall and Auchrannie Spa and Country House, in Brodick, are open for residents with power, heat and refreshments, North Ayrshire Council advised.
Lamlash fire station is open all day with power available through a local generator.
Other areas badly affected by power cuts are around Campbelltown and on the islands of Islay, Jura and parts of Bute, according to Argyll and Bute Council.
"Contingency arrangements are in place to make sure that people have food, heat and are well cared for," the council said in a statement.
"Everyone is working closely together to cope with the impact of the severe weather in Argyll and Bute communities and local companies have provided support and equipment.
"We would encourage people to look out for older or vulnerable people in their communities, especially elderly relatives, friends and neighbours."
More than 14,000 properties suffered power cuts in Dumfries and Galloway, Argyll and Arran yesterday.
A spokesman for energy provider ScottishPower said: "We have approximately 6,000 homes still off supply in the south-west, predominantly in the Wigtownshire area.
"We are working closely with the local authority who are seeking to clear roads of snow to allow us to access the faults. The main issues are still blocked roads and snow drifts."
Driving conditions were also widely affected by snow and high winds, mainly in the south-west and north-east.
The A83 was closed at Clachan, Kintyre, after yesterday's heavy snow where excavators are helping to clear the blockage.
The A711 was closed at Dalbeattie, as were the A701 Moffat to Devil's Beeftub and A77 Portpatrick to Stranraer.
Colin MacKenzie, of trunk road operators Amey, said: "The severe snowfall throughout Friday and overnight into Saturday morning has resulted in treacherous driving conditions across various routes on the south-west trunk road network.
"Amey has deployed additional crews who are working around the clock to clear the A77 at Ballantrae. However, we are battling against relentless snowfall and high winds resulting in snow drifts of up to four feet. The route will remain closed until further notice.
"Amey will continue to concentrate crews on affected areas to ensure roads are kept open wherever possible. Although all routes are treated, we cannot guarantee they are ice free."
Further "significant" snow fall is expected to accumulate on the hills, forecasters said.
Snow showers affected the Grampian region, where some roads were closed, particularly around Deeside and further north-east.
Snow gates were shut in the morning at the A939 Cock Bridge-Tomintoul road while the Ballater-Corgarff stretch off the road was also closed due to drifting snow.
Gates were also closed at the A93 southbound at Braemar.
Transport Minister Keith Brown said Scotland's 130-strong fleet of gritting and road clearing vehicles would be working "flat out" throughout the weekend.
He urged drivers to heed police advice and plan any journeys in advance.
The extreme weather conditions prompted a meeting of the Scottish Government's Resilience Room last night. Mr Brown, after chairing a further "resilience" meeting today, said more than 18,000 homes are without power.
"While the worst of the snow is slowly abating, high winds blowing snow already on the ground continues to make clearing the A75 and A77 in some sections difficult," he said.
"Every effort is being made collectively by our road operating companies, local authorities and other partners to clear the main routes. Some drivers chose to stay in their cars overnight on the A75 and they were supported by emergency services and the local mountain rescue volunteers. We're not aware of anyone who was not offered a form of accommodation last night.
"I'd like to pay tribute to the efforts of all involved. From council employees working flat out to support the vulnerable and those in rest centres through the dedicated staff in all our emergency services and utility companies who worked through the night we've seen a joined up and committed response.
"I'd particular single out those communities and volunteers who have contributed - the great work done by Mountain Rescue and by farmers in clearing roads and supporting the needy in Dumfries and Galloway are just some examples.
"While we're doing all we can to minimise travel related disruption, we'd urge drivers to follow police advice, particularly where the advice is not to travel. Please plan all journeys in advance and use all the available traffic and travel information."
"The Scottish Government's Resilience team will continue to co-ordinate response and recovery activity throughout the weekend and offer assistance where possible."
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