Motorists are asked to drive carefully this morning with road temperatures just below zero in many places but particularly in the northwest and the midlands.
There is also some frost in Munster.
There are icy conditions between Youghal Bridge and Rincrew and around the Youghal bypass. There have been several crashes in this area this morning.
Extra care is advised on high ground in Donegal especially on the Glenties/Letterkenny Road over Meenaroy, around Glengesh, through the Glenshane Pass and around Binbane.
It is icy on secondary roads around Cavan town and in sheltered areas around Monaghan town.
Black ice has been reported around Navan and gardaí are reporting very icy conditions around Dundalk.
It is very icy on the Roscommon/Athlone Road. Extra care is also needed on the Kilbeggan/Mullingar Road.
Met Eireann has predicted snow for high-lying areas in the north and northwest.
The first snow flurries were reported at Knock and Belmullet, Co Mayo, on Sunday. Parts of Ulster and Connacht were also covered yesterday morning.
“Winter has really kicked in,” said Met Éireann forecaster John Eagleton. “November was really autumn; there has been a noticeable change now in December.”
However the unusually cold weather experienced last winter is unlikely, with winds coming in from the Atlantic Ocean rather than blowing down from Siberia, as occurred 12 months ago.
A white Christmas on the scale of last year seems unlikely, although Met Éireann do not make weather forecasts that far into the future. Things will warm up later in the week, with daytime temperatures on Thursday expected to rise to as high as 11 degrees.
However, for areas on higher ground, particularly in the north and northwest, more snowfall remains likely.
Brendan Flanagan of Inishowen, Co Donegal, and a former employee of the national forecaster, said local people are more than well prepared for the “dusting of snow” they have received so far.
After some local amateur forecasters predicted heavy snow at the end of October, shops in the area experienced a major boost in the sale of winter items such as snow tyres, he said.
The appropriately named Slieve Sneacht was one of the first areas to be spotted with snow, according to Mr Flanagan.