The stormy weather has also resulted in ferries being cancelled and delays to rail services.
A number of trees came down overnight in Aberdeen, causing roads in the city centre to close, including Union Terrace and Willowbank Road.
The city council said it received dozens of calls about fallen trees and crews are working to remove them.
Drivers were also warned to take extra care on the Esplanade after sand and water was blown on to the road.
In Dundee a driver suffered minor injuries after a tree came down and hit a car on Arbroath Road near Baxter Park at about 9am.
Fife Constabulary also reported trees down in the area, with "Standing Stane road" between Kirkcaldy and Leven shut as well as the road from Cupar to Melville Lodges roundabout.
Victoria Kettley, a forecaster with Meteogroup, said: "The strongest winds at the moment are through Perth and Aberdeenshire, with gusts of up to 72mph locally.
"Through the course of the day, the wind will shift westwards.
"This afternoon, winds down the west coast could gust at 50mph to 60mph, with the main zone being on the western fringes to the western isles, easing overnight.
"Gusts on the east coast will ease rapidly over the next couple of hours."
High winds also led to restrictions on bridges, with the Forth Road bridge opened to cars only this morning.
On the railways, ScotRail reported "minor disruption" to its West Highland line.
For those hoping to travel by ferry, many Caledonian MacBrayne services were facing disruption and some routes were cancelled.
Flood warnings are in place for Haddington in East Lothian and many parts of the Borders.
A further nine flood alerts were issued across Scotland, including in Aberdeenshire, Dumfries and Galloway and Fife.
The Met Office put out yellow "be aware" warnings of severe weather for much of the country and forecasters predicted the most persistent rain would be in the south and east throughout the day.
Transport minister Keith Brown is leading the Scottish Government's resilience committee meeting this morning to receive updates on the transport network, utilities and flooding.
The resilience committee meeting was told that about 2,000 households across a number of areas were without power, but electricity providers had staff trying to reconnect customers as quickly as possible.
It also heard the heavy rain had seen some areas record about 40mm of water in the past 36 hours.
Mr Brown said: "Yet again, we have seen a wide range of agencies and organisations react quickly to weather alerts and to put in place their response plans for the potential impact of heavy rain and high winds.
"This activity, including a significant level of multi-agency co-operation, has played its part in keeping disruption to a minimum in difficult conditions.
"The travelling public also deserve praise for the way they have reacted to the various travel updates this morning.
"But no-one is being complacent and whilst the weather alerts remain in place, the focused response will continue."
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