Edinburgh Airport has reopened after heavy snow which has also caused problems on the roads.
The airport shut at 0915 GMT, but was reopened at about 1800 GMT. Passengers have been told to contact their airline.
The airport was closed for three days last week due to heavy snow.
The Forth Road Bridge has also reopened after it was closed southbound. There are long delays on the M8 and city bypass.
There were reports that traffic heading away from Edinburgh Airport was at a "virtual standstill" with "very long delays".
Lothian Buses said all its buses had been suspended.
A passenger said it took two hours on the Edinburgh Airport bus to get back into central Edinburgh just before buses were suspended.
Heavy snow in the Dunfermline/Cowdenbeath area has led to Stagecoach East Scotland stopping all its buses, apart from service 19.
A spokesperson for Stagecoach East Scotland said: "Bus services are continuing to be disrupted by the severe weather conditions, which is causing delays and cancellations to services.
"Many services across Fife are currently suspended due to the road conditions and blockages along the routes.
"We are doing everything we can to operate as full a network as possible. However, safety is our absolute priority and a number of services are suspended at present until we feel it is safe to operate.
"We will continue to monitor the weather situation closely and take decisions based on the road conditions to ensure the safety of our passengers, staff and other road users."
First Buses has suspended all its services in West Lothian due to the weather.
Edinburgh Castle is closed due to the weather conditions.
Edinburgh City Council said most of its schools had reopened on Monday, but a decision would be made later about whether they would be open on Tuesday.
Some schools in Fife are open, although many which had been preparing to open on Monday have been closed.
Bryan Kirkcaldy, Fife Council's head of education, said: "In the last couple of hours the weather has turned ferocious, with blizzard conditions in places and new snow falling on icy roads, making all travel very dangerous.
"While it is no-one's wish to close schools at such short notice, the change in conditions this morning is making travel difficult throughout the kingdom and in the interests of safety, schools which were previously open this morning are now starting to close.
"I'd urge all parents to keep up to date on Fifedirect, Twitter, Facebook and listen in to local radio for regular updates on schools which may now be closing.
"We will do our best to get the information to parents as fast as possible and we will keep updating regularly as the day progresses."
Most schools in East Lothian schools are open.
Lothian and Borders Police are advising motorists against all non-essential travel.
There have been a number of reports of vehicles becoming stuck on roads and motorways.
A police spokesman said: "If people have to drive, then we advise that they make sure their car is fit for travel and well equipped for the weather.
"Likewise people should be wary of the road conditions, and exercise caution behind the wheel."
The force also urged motorists to be vigilant after a recent spate of weather-related vehicle thefts in Edinburgh.
At about 0845 GMT on Sunday a 50-year-old woman was de-icing her Hyundai Getz car in Stenhouse Street West when a man in his 20s wearing tracksuit bottoms jumped into the driver's seat and drove off.
The car was recovered a short time later in Stenhouse Drive.
On Friday, a 24-year-old man was driving in the area at about 1830 GMT when a group of youths threw snowballs at his silver Ford Focus.
After he got out to remonstrate with the group, one of them jumped into the car and drove off. The vehicle was recovered around an hour later in Quality Street Lane.
All Midlothian schools are staying open on Monday afternoon.
A Midlothian Council spokesman said: "The forecast is for current snow showers to ease in the area, and it is possible bus services may resume in the next couple of hours.
"Parents who wish to take children from school early may do so.
"Arrangements are being made for schools to stay open longer than normal to ensure children are looked after until parents or carers can arrive to pick them up."
The council said all schools on Tuesday would be open, but urged parents to keep checking for updates.
However in West Lothian all schools and nurseries will be closed on Tuesday.
All East Lothian schools will be open on Tuesday with late closures being announced on the council website.
Fife Council said it would be the decision of each head teacher whether their school reopened or remained closed on Tuesday. The Fife Council website will be updated at 0700 GMT on Tuesday.
Most Edinburgh schools are to open on Tuesday but check Edinburgh City Council website in the evening because there will be some closures. All special schools will be closed.
Edinburgh Napier University has cancelled all exams that were due to start at 1730 GMT and all exams on Tuesday.