Forecasters said temperatures could dip to -4°C across the next few nights in Glasgow, while in rural spots it could fall to -9°C.
During the day the temperature is expected to hit 3°C.
But the Met Office has issued yellow weather warnings for some parts of Strathclyde, as well as the Lothians, Borders, Grampian, Central, Tayside and Fife and the Highlands, with rain and sleet mixing with snowfall in some areas.
Forecaster Peter Sloss said: "Glasgow has escaped the snow, but it's still cold.
"It is expected to be a dry week with very few showers. If there are showers they will only be wintry at higher levels in west Scotland."
The AA warned that 75% of drivers were not prepared for conditions on the roads, and urged drivers to carry an essential winter kit and check their cars before getting behind the wheel.
The RAC is expecting up to 56,000 breakdowns and widespread disruption across the UK.
It placed extra patrols on standby to help stranded motorists and said call-outs were likely to rise 20% or more.
Scottish Transport Minister Keith Brown said councils had increased amounts of grit and salt to treat major routes.
While some drivers face transport problems, others were enjoying the snow yesterday.
Children in Lanarkshire and the Borders took the opportunity to build snowmen, have a snowball fight and even cycle through the snow.
Online retailer Amazon said its sale of sledges in the UK surged 600%, while Tesco said its sales of shovels were up more than five times last week's levels.
Car products such as windscreen wash, anti-freeze and ice scrapers are already 25% higher than last week.
Glasgow has escaped the snow, but it is still cold