The severe weather is expected to last until today with forecasters warning people to prepare for more snow.
More than 1,000 pupils were affected in the Highlands and Moray areas by the cold snap.
Road conditions and loss of heating led to councils keeping some schools closed for the day.
School closures in the Highlands included Aviemore primary, Carrbridge primary and its nursery, Stratherrick primary, Daviot Primary and Kingussie High.
In Moray, Speyside High was closed due to adverse weather conditions with pupils advised on the council’s website “do not wait any longer than 15 minutes at bus stops.
“If your bus has not arrived in that time pupils should return home.”
Aviemore, Kingussie and Speyside were among the worst affected areas.
Central Scotland had also been put on alert after a frosty weekend but the region escaped major disruption as rain fell instead of snow yesterday.
Last night a spokeswoman for the Met Office said: “There has been snow overnight in various areas but conditions will be improving over the next few days. We are not expecting more snow with the only snow being on high ground. The week ahead will be milder than average.
“While rainfall totals are not expected to be particularly high, in view of current ground conditions be aware of the possibility of some localised flooding.
“Outbreaks of rain, heavy at times, reaching south-western England on Tuesday morning will move fairly quickly north-eastwards across England, Wales and Northern Ireland to lie over a large part of Scotland by Tuesday evening. This rain is likely to last approximately four or five hours in any one place.”
Traffic Scotland said that while no trunk roads had closed the B974 Banchory to Fettercairn was shut at the snow gates. Snow has also affected other roads including the A95 and A96 at the Glens of Foudland. The Ballater to Corgarff road was closed in both directions.
The transport agency also advised drivers to use caution in the icy conditions.
The coldest temperature in Scotland on Sunday night was recorded at Sutherland with the mercury falling to -6.7C.
However, the snow has provided a boost to ski resorts.
Ross Coulter, spokesman for CairnGorm Mountain, said the 20cms of snow which fell yesterday had attracted a lot of interest in the resort.
Mr Coulter said: “We certainly see a boost on social media channels when it snows, it builds a lot of excitement on sites like Facebook and Twitter. We’ve had our snow cams on and we’re hoping the snow will last.
“However, we also tell people to be careful and try to change their awareness of the winter conditions.”
Areas above 200 metres will be most impacted, but land up to about 150 metres can still expect up to 5cm of snow.
The Central, Tayside, Fife, Grampian, Highlands, Eilean Siar and Strathclyde districts are all subject to a severe weather warning today.
The forecast warned: “Some icy patches from rain falling on to sub-zero surfaces are also likely this evening. Please be aware of the possibility of localised disruption to travel.”
The AA said a “distinct north-south divide” in temperatures triggered a series of breakdowns at the weekend, with drivers in North Wales and northern England contending with freezing temperatures.
Spokesman Max Holdstock said: “It’s difficult to know exactly where the rain will fall as snow but the weather is changeable, with heavy rain and sleet with a risk of snow accumulating on higher ground.
“This has led to some difficult driving conditions at times and the risk of further localised flooding.”
Mr Holdstock urged drivers to allow more distance between them and the vehicle in front, moderate their speed and anticipate delays.