Met Éireann has warned that overnight temperatures could drop as low as -6 degrees celsius in the days ahead.
The forecaster has issued a status yellow weather warning for snow and ice for the entire country for the coming week.
People are being warned of heavy showers in some areas, with accumulations of snow likely on higher ground. There will be widespread frost at night with the potential for icy conditions on untreated surfaces as sub-zero temperatures return.
Met Éireann meteorologist Matthew Martin said that the cold weather is expected to last until at least Friday.
"It looks like we're in for one of the coldest weeks of the winter season so far," said Mr Martin. "Some quite cold air is going to funnel in across Ireland from Tuesday onwards and we're expecting some wintery showers and widespread frost at night."
Tonight is set to be very cold and mainly dry with clear spells. The west of the country is likely to see heavy showers, with falls of sleet and snow further inland. There will also be widespread sharp or severe frost and icy patches. Temperatures could drop as low as -3 in some parts.
Wednesday will continue to be very cold, though there will be some sunny spells. The weather looks dryer and calmer, with temperatures in the range of 3 to 5 degrees.
However, Met Éireann has forecasted a significant drop overnight on Wednesday and in the early hours of Thursday with temperatures set to hit as low as -4 to -6 degrees in parts of the country. Cold, stagnant air will run up against a low-pressure system, increasing the likelihood of snow in some areas too.
Frost and ice will develop on Thursday night along with further wintry showers. Friday will continue very cold with sunshine and further wintry showers. Highest temperatures of 3 to 6 degrees with moderate to fresh northerly breezes.
Next weekend will continue very cold with sunny spells and wintry showers. There will be frost and ice at night.
The weather warning for the coming days follows a weekend of turbulent weather in some areas. Galeforce winds hit parts of the country on Saturday and Sunday, causing power outages and fallen trees which blocked roads and damaged properties.