There has been damage to property but no injuries. Forecasters have indicated that it will be dry and bright for most of Scotland by Wednesday afternoon.
Ferry services have been considerably disrupted and falling trees have caused some problems on roads and railways.
Transport Minister Keith Brown chaired a meeting of the Scottish government's resilience team.
Mr Brown said: "No-one is being complacent and whilst the weather alerts remain in place, the focused response will continue."
A train travelling south from Aberdeen to Edinburgh suffered minor damage when it hit some fallen branches and a bus was hit in Aberdeen.
In Edinburgh a tree landed on the side of a house in Redford Road. No-one was injured. High winds has also forced the closure of the Royal Botanic Garden in the capital.
Aberdeen City Council said trees were down across the city, and the tree squad was working flat out.
Huge quantities of foam blown in from the sea covered cars, buildings and a park beside Aberdeen beach at the Footdee area.
Scottish Borders Council activated its emergency response team after warnings of extremely heavy rain, while Tayside Police warned of an increased risk of surface water on many routes and possible river flooding.
Part of a building roof has come down blocking Bridge Street at Peterhead harbour and Grampian Police said minor roads across the region have been blocked by trees.
High winds have seen restrictions on some of Scotland's bridges, while fallen trees are affecting some minor routes and ferry crossings.
Historic Scotland has shut several of its sites across the country including: Fort George, Craigmillar Castle, Loch Leven castle, Inchcolm Abbey, Tantallon Castle, Seton Collegiate Church, Tolquhon Castle, Kisimul Castle, Spynie Palace, Elgin Cathedral and Inchmahone Priory.
The National Trust for Scotland has closed: Brodie Castle, Newhailes, Pitmedden, Preston Mill, Fyffie castle, Haddo House, Castle Fraser, Leith Hall Gardens, Crathes Castle, Craigevar Castle, Drum Castle and Crarae Gardens.
Heavy rain and severe winds have caused power cuts and travel problems and swept sea foam into parts of Scotland.