A number of schools were closed on Monday as snow and ice swept over the country on Sunday night and Monday morning.
A status yellow weather warning for snow and ice was issued on Monday and is effective from at 3am on Tuesday to 10am on Tuesday.
Met Éireann said air temperatures will fall to freezing or below later on Monday night and early on Tuesday morning, with widespread frost and ice on untreated surfaces.
Scattered wintry showers are also expected, mainly in west Ulster, Connacht and west Munster.
A second status yellow weather warning is in place for the entire country, effective from 6pm on Monday until 6am on Tuesday.
The national forecaster said a spell of wet and windy weather, with thundery downpours, will sweep eastwards over the country during Monday evening and night.
Gusts of between 90 and 110km/h will occur in Atlantic, southern and later eastern coastal fringes for a time.
Some accumulations of snow are possible during this period also.
AA Roadwatch has advised drivers to slow down and remember that stopping distances are up to ten times longer on icy roads.