Staff from Scottish Power Energy Networks hope to restore power to most properties on Monday - but some homes could have to wait longer.
Drumlanrig Castle and RSPB Mersehead are urging people to stay safe with red and amber weather warnings set to come into effect.
Two red weather warnings are in place as winds of up to 100mph are forecast to hit Northern Ireland and Scotland.
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Storm Éowyn is battering the west coast as a rare red weather warning has come into effect in Scotland. The area covered by the high winds alert - which lasts until 17:00 - has been extended to include most of Dumfries and Galloway. It already covered most of the central belt, including Glasgow, Edinburgh and some islands.
Disruption is continuing after Storm Eowyn battered Scotland, claiming one life and leaving thousands without power. On Sunday evening around 16,000 properties were still without electricity as power companies worked to restore supplies after the storm on Friday.
A rare red weather warning has come into effect in Scotland as Storm Éowyn causes widespread disruption. The area covered by the high winds alert - which lasts until 17:00 - has been extended to include most of Dumfries and Galloway. It already covered most of the central belt, including Glasgow, Edinburgh and some islands.
A Dumfries and Galloway family will feature on the new series of a hit BBC Scotland TV show. The Thomsons, who farm near Balmaclellan, appear on the first episode of the latest run of This Farming Life on Tuesday.
The Met Office issued a red weather warning for Friday as very strong winds are expected to disrupt the country.
Follow here for all updates including traffic disruption and closures as Scotland is battered by winds forecast to gust up to 100 mph with a rare red extreme weather warning issued.
Shops including Asda, Sainburys and Tescos have closed across Scotland and Ireland on Friday due to dangerous weather conditions.
A red danger to life warning for wind has been extended in Scotland where more than 35,000 properties have lost power and people have been urged to stay indoors. Trains have been suspended across the country and hundreds of schools are closed after forecasters predicted wind speeds of up to 100mph.