The M90 Queensferry Crossing and the A90 Forth Road Bridge are currently closed to high-sided vehicles, though cars are still allowed to cross, and the Tay Road Bridge is closed to all vehicles.
A football fixture between Aberdeen FC and Johnstone FC was due to take place this afternoon but the risk posed by the wind has forced a cancellation.
Following this Police Scotland advised people to avoid travelling unless absolutely necessary.
The Met Office issued an Amber Weather warning for wind at 7am covering the east coast which is due to lift at 3pm. A yellow wind warning is set to remain in place until midday on Monday.
A statement on its website said: “An area of very strong west or northwest winds will cross Scotland and northeast England on Saturday morning in association with Storm Malik before easing during the afternoon.
"Gusts of widely 50-60 mph are likely with a short period of gusts in excess of 75 mph possible, particularly for Moray and north Aberdeenshire as well as the Lothians.”
ScotRail has reported a lot of disruption this morning with a number of fallen trees blocking tracks and causing closures and delays. It is understood that there have been almost 300 cancellations. A ScotRail spokesperson said: “The impact of Storm Malik has affected many parts of Scotland’s Railway. Some train services are experiencing cancellations or delays as trees and other debris have been blown onto the tracks.
“We’re advising customers who need to travel to check their journey on our app or website before they travel.”
Network Rail confirmed that until the storm has passed and lines have been inspected for damage, all train services are suspended between Dundee and Aberdeen, Aberdeen and Inverness, Inverness and Kyle/Wick/Thurso and on the West Highland line north of Helensburgh.
Speed restrictions also remain in place on the East Coast Mainline, North Berwick branch line, and Borders Railway.
For a period on Saturday morning the line was also closed at Kirkcaldy after a trampoline was blown onto it.
Liam Sumpter, Network Rail route director for Scotland, said: “We have taken the decision to stop trains running in some parts of the country while we assess the impact of Storm Malik on our infrastructure.
“The safety of our customers and colleagues is always our first consideration during severe weather.
“We will reopen the affected lines as quickly as we can for our customers, but are urging passengers to check their journey before travelling.”
Elsewhere it has been reported that the roof of a salon in Blackburn has blown off onto Bathgate Road with people in the area being urged to be careful as there is a possibility of more debris falling.
Similarly, there has also been a report that the roof has blown off the Lanark Leisure Centre.
One Edinburgh resident, from Abbeyhill, shared pictures of a fallen tree blocking the entrance to her flat, while another image shared with the Scotsman shows the scene of a tree in Dick Place completely blocking the road.
Fife residents have reported several fallen trees blocking various parts of the A92 including near Ladybank and between Kirkcaldy and Glenrothes.