Up to 20cm of snow is forecast to fall over the east of the country in the coming two days - a record for the month of November.
The big freeze is set to continue after a severe weather warning for heavy snowfall and overnight temperatures fell to as low as minus 10C.
Transport Minister Noel Dempsey said main roads are being treated and salted, but advised motorists not to make any unnecessary journeys and to use public transport.
"The cold snap is set to continue until at least the weekend and will probably go in to next week," Mr Dempsey said.
While five to 10cm of snow is expected to fall in inland locations by Tuesday evening, residents in counties Louth and Dublin are bracing themselves for traffic chaos, where between 10 to 20cm will settle.
Up to 25cm will fall along the foothills of the Wicklow mountains. A previous November high of 12cm was recorded in 1977.
Meteorologist Pat Clarke, of Met Eireann, said there will be frost every night, with the added complication of hail, sleet and snow.
"There are no signs of any mild Atlantic-type weather systems that will bring us reasonable temperatures and rain in from the Atlantic that would help us clear up this frosty weather," said Mr Clarke.
Michael Egan, of the National Roads Authority (NRA), said adequate salt stocks are being distributed around the country but warned local authorities have to prioritise routes. Councils along the east coast have already used 2,000 tonnes salt a day over the weekend - three times more than normal - in a bid to keep roads clear.
"We have in excess of 90,000km of public road in the country," Mr Egan said. "The national road network accounts for just over 5,500km of that, so the balance of what local authorities can do has to be restricted."