The Met Office has a yellow warning in place for ice which covers large parts of southern and eastern Scotland until late morning.
Forecasters warned of ice forming on roads and pavements, increasing the risk of accidents, slips and falls.
Bitterly cold temperatures will continue as the wintry snap dubbed the “mini beast from the east” keeps its grip on the UK.
There were a small number of cancellations at Edinburgh Airport today after snow affected flights elsewhere in the UK.
The airport said that flights to and from Heathrow and Bristol, where the runway was closed Sunday morning, were affected.
Five arrivals and five departures were cancelled yesterday.
In a tweet the airport said: “There have been a small number of cancellations today due to the weather. If you are due to fly today please contact your airline for the latest information on your flight.
“If you are travelling to the airport please leave extra time for your journey.”
Meanwhile, a hillwalker was rescued in challenging weather conditions on Saturday night after getting into difficulty on The Cobbler in Argyll.
Arrochar Mountain Rescue Team said the woman – who was taken off the hill on a stretcher – was thought to have broken an ankle.
Temperatures could get down to as low as -2C or -3C in Edinburgh and Glasgow Monday night into Tuesday, with the wind chill making it feel considerably colder.
Across the country as a whole, temperatures are expected to climb later in the week, bringing widespread rain.
Met Office meteorologist Mark Wilson said: “It’s going to be a very, very cold start, with a widespread frost and ice around as well.
“It will stay windy, and it will stay very cold indeed.
“Temperatures barely above freezing, and in the wind it will feel like it’s well below freezing.”
The weather warnings cover parts of central Scotland as well as Tayside and Fife, Grampian, the Highlands and western isles, southern Scotland and Strathclyde.
However, Scotland looks set to avoid heavy snowfalls forecast for parts of England and Wales.
Mr Wilson said people could see 10-15cm of snow, and up to 25cm over higher ground south of the Border.
He said snow showers in other parts of the country are expected to ease throughout the day, with some spells of sunshine possible.
Heavy snowfall and bad weather in Ireland yesterday led to the cancellation of a number of events, including the homecoming party at the Aviva Stadium for the country’s grand slam-winning rugby team.