They have also warned of unsafe driving conditions, which are expected to persist tonight, Monday, March 20, and in the morning, Tuesday, March 21.
Met Eireann is forecasting cold conditions overnight and into tomorrow with some snow accumulating even at lower levels.
He said: "On Monday a deep low pressure centre drifted across the far north of the country bringing strong and gusty winds and has helped to drag cold air in from the Arctic".
At least 1cm is likely to fall in most places, with more than 5cm predicted on higher ground. "There's been very strong storm-force winds that will be clearing northeast-wards".
The Met Office said it was not unusual to see snow at the start of spring.
Showers are expected to turn to sleet and snow, with showers being heaviest and most frequent in the western half of the country.
Londoners will wake up to temperatures of 4C or 5C on Tuesday morning with the temperature expected to rise to 9C by the middle of the day.
Drivers in the north-westerly regions that have been told to take the most care as snow and ice could play havoc with road surfaces.
There will be clearer skies as the week goes on, with daytime temperatures warming up to 12C on Friday, with highs of 14C forecast for Saturday.
Temperatures will range from 4 to 7 degrees but windy conditions in the morning will make it seem much colder.