Yellow "be aware" warnings are also in place for snow in southern and central Scotland and ice in the north east.
The amber warnings cover the Borders, Midlothian and East Lothian and the Shetland Islands.
About three to six centimetres (one to two inches) of snow was widely expected with 10cm (4in) possible on higher ground.
The warnings are valid overnight from Wednesday into Thursday.
Heavy snow started to fall across central Scotland early on Wednesday evening.
Bridge of Allan and Stirling were among the first places to be affected.
Scottish Borders Council has opened its emergency planning centre which was expected to remain operational until at least 10:00 on Thursday.
Ch Insp Andy McLean said: "We are urging members of the public not to travel unless their journey is absolutely necessary from 6pm tonight until 9am tomorrow.
"If we get the levels of snow forecast then travel across the region will become extremely difficult, and high volumes of traffic will cause issues with snow clearance, as was the case in the Hawick area last Friday when we had sudden, heavy snowfall.
"If any motorists become stranded we would advise them to contact Police Scotland on 101 and wait for assistance. Drivers should not abandon their cars, as this leads to further problems."
Gritters have been pre-salting primary routes and will be running throughout the night.
Additional resources have also been put in place on a number of key routes by Transport Scotland, including the A68 at Soutra and Carter Bar and on the A7.
As a result of the snowfall forecast, and the likely levels in the morning, the council has taken the decision to cancel all school transport on Thursday morning.
A decision on school closures was expected by 06:00 on Thursday.
Jim Fraser, SBC's emergency planning officer, said: "We are urging members of the public to stay off the roads tonight if at all possible and keep up to date with the latest situation via local radio and the council's website and social media channels.
"All necessary winter maintenance has been carried out this afternoon on primary routes and our gritters will be out throughout this evening in an attempt to keep key routes open for emergency travel."
The Met Office has issued an amber warning for overnight snow in south east Scotland and the Shetland Islands.