A Met Office statement said: "Frequent and heavy hail and snow showers are likely to affect many parts of Scotland and Northern Ireland from Monday evening, continuing for much of the day on Tuesday". Some patches of fog will form and also some frost.
One forecaster warned that a "potent wintry blast" is on its way and could arrive in the United Kingdom "early next week".
Up to 2in of snow could fall at lower levels, with up to 4in predicted in areas above 656ft.
A woman was spotted pushing a pram on bitterly cold and frosty morning in Aberystwyth, Wales, on Thursday.
The warning, which was originally issued at 2am this morning, now runs until 2am Friday morning and is in effect for the entire country.
Nearly the whole of Northern Ireland, and the west of England, including Liverpool, Manchester, Wolverhampton, Bristol and parts of Devon will also be affected by thick fog.
Forecaster Helen Roberts said: "Mists will quickly develop tonight turning readily into fog, mostly across western areas".More news: 'X-Men' Spinoff 'Gambit' Loses Director Gore Verbinski
SNOW and ice could hit West Fife on Monday and Tuesday.
The gardai also warned motorists to take care this morning.
The Met Office has issued a weather warning for heavy hail and snow in Dundee, Tayside and Fife next week.
'We have already seen fog around and it will continue over the next couple of hours and into today.
"In addition some roads and pavements may become icy, increasing the chance of injuries from slips and falls".
Met Éireann say that the fog will be most dense inland, away from coastal areas, and will lead to poor driving conditions, so if you are going to be on the road, you might want to allow a little extra time to get from A to B.
"Slower journey times are possible and there is a chance of delays to flights".