It follows significant rail disruption after a busy line between Edinburgh and Glasgow was closed for hours because of a signalling fault.
Dozens of trains on the line via Bathgate and Airdrie were cancelled or delayed.
Police Scotland said areas likely to be affected by wintry conditions included the Borders, East Lothian, Aberdeenshire, and much of the Highlands and Islands including Orkney and Shetland.
Inspector David Hynd said: “Warnings from the Met Office indicate icy patches can be expected; and lying snow of between 1 and 3m could also be experienced.
"Drivers should therefore exercise EXTRA CAUTION while this is in force, which is between 6pm tonight and 12 noon tomorrow.
“If you are travelling, you should ensure you and your vehicle are adequately prepared for the conditions, making sure you have sufficient fuel and supplies such as warm clothing, food and water in the event you are delayed for several hours.
"Charge your mobile phone and plan your route, as well as alternative routes."
Fog caused a further hazard for drivers on the Queensferry Crossing over the Forth between Fife and Edinburgh.
Temperatures are expected to plunge again to -10C in Braemar, -9C in Aviemore, -5C in Glasgow and -4C in Edinburgh.
BBC Scotland Weather tweeted it had fallen to -14.4C in Braemar last night - the lowest UK temperature for seven years
ScotRail said the cause of the signalling fault at Caldercruix in North Lanarkshire was being investigated, with engineers en route.
It said Milngavie-Edinburgh on the line had been suspended.
Helensburgh-Edinburgh services were being terminated/started from Airdrie.
Replacement buses have been arranged for Balloch, Dumbarton Central and Airdrie.
Passengers could also use their train tickets on First Bus services between Bathgate and Edinburgh.
Further disruption was caused by a train breaking down at Hyndland, in the west end of Glasgow, at one of the busiest points on the Scottish network.
ScotRail said the train broke down due to a loss of air, which disrupted services through Glasgow Central and Queen Street low level stations.
It said the train was back at the depot with the cause of the fault under investigation.
A spokesperson for the ScotRail Alliance, which includes track and signalling body Network Rail, said: “We’re sorry to customers who experienced disruption to their journey this morning.
"Anyone delayed by 30 minutes or more is entitled to claim money back through our Delay Repay guarantee.
"Just keep hold of your ticket and claim on the ScotRail website or mobile app.”