Update 1.30pm: A status yellow snow-ice warning has been issued for 10 counties.
The warning will affect all of Connacht and counties Cavan, Monaghan, Donegal, Longford and Louth.
It will come into effect from just after midnight and last until noon on Wednesday.
Met Eireann says there will be icy patches, while hail and sleet will fall mainly on higher ground.
Status Yellow Warning issued for Connacht, Cavan, Monaghan, Donegal, Longford and Louth.
Valid from 00:00 hours Tue, 02-Apr-2019 to 12:00 hours Wed, 03-Apr-2019 pic.twitter.com/fZClWXkBz4— Met Éireann (@MetEireann) April 1, 2019
Update 8.52am: There is to be a dramatic drop in temperature this week compared to last week's high of 17 degrees.
Met Eireann has warned tonight will be very cold.
There will be night-time lows of minus 1 this week, with some wintry showers expected.
Forecaster Vincent O'Shea says some areas may even see snow.
"Upland areas of Ulster and Connacht especially would be the areas most likely to get a wintry precipitation, probably rain, sleet and snow, probably hail as well.
The forecaster says it will be cold and unsettled for the rest of the week, with a mix of bright spells and showers, some of them wintry.
However, good weather is set to return at the weekend.
Early indications suggest that temperatures will return close to normal and there will be a good deal of dry weather, with sunny spells, but with some further scattered showers too.