Met Éireann says that the Southwest and parts of the north and east coasts remain at risk of further snow showers over the next 24 hours.
Rebecca Horan from AA Roadwatch says sub-zero road temperatures are being experienced throughout the country.
Speaking after a meeting of the Government's emergency taskforce earlier today, forecaster Joan Blackburn said milder temperatures would arrive around Christmas Eve, bringing a thaw.
She said it would remain dry in many areas over the coming days, but there is a risk of some snow on Tuesday in Munster and parts of Leinster.
Sean O'Neill from the National Roads Authority said national supplies of salt were down to 10,000 tonnes, with 2,000 tonnes a day being used.
He said the Authority is now confident that the new shipments at sea would begin arriving into Cork and Belfast around Wednesday, and a fleet of 130 trucks would be on hand in Cork to distribute the supplies to the counties most in need.
He said the priority counties now are Monaghan, Donegal, Cavan, Roscommon, Kerry and Cork.
Inspector John Ferris from the Garda Press Office said traffic units around the country had reported that many people are still driving too fast and too close to the car in front for the prevailing conditions.
He appealed to people to drive at an appropriate speed.
Committee chairman Sean Hogan asked the public to take care when out and about and repeated advice that people walking on icy footpaths should avoid walking with their hands in their pockets and while talking on a mobile phone.
However, he suggested pedestrians take a mobile phone with them when walking in case they need assistance.
He also asked people to stay away from iced over water.
Motorists warned of dangerous roads
Motorists are being urged to take extreme care on icy roads today, with temperatures below zero in many areas.
AA Roadwatch says the worst affected areas are west Munster, Connacht, and north Leinster.
Dublin, Belfast, Shannon and Cork airports are open, but many flights to London Heathrow and other European airports have been cancelled due to airport closures.
A spokeswoman for the Dublin Airport Authority said that there had been a light snow shower overnight with temperatures as low as -9C, but ice teams were out spraying the runway overnight.
Across Europe the severe winter conditions have caused chaos at airports - with flights affected in the UK, Germany, France, Italy and the Netherlands.
London's Heathrow Airport will not be accepting any arrivals today and will only allow what it described as a 'handful' of departures.
Staff at the airport are working to clear snow and ice in order to permit a full reopening of the airport tomorrow.
Gatwick, where aircraft were also grounded yesterday, has reopened, although widespread disruption is expected.
Passengers are advised to contact their airlines by telephone or online before travelling to the airports.
Most Bus Éireann services around the country are operating as normal, with some delays and diversions in affected areas.
Services currently affected by weather and road conditions include some in the south, southwest and northwest.
Irish Rail is expecting a full service on all Intercity, DART and Commuter routes. Dublin Bus services are operating with some curtailments.
No reported overnight snowfalls in Cork or Kerry
Despite forecasts, there was no snow in Kerry overnight.
Gardaí at Cahirsiveen in the south of the county said it was raining there this morning, and a partial thaw was setting in.
However, roads there and throughout the rest of the county continue to be extremely dangerous.
Some of the worst affected roads are in housing estates in towns and villages, as well as secondary roads in rural areas, as these have not been salted or gritted.
There were no reports of snow overnight in Cork City and county, where temperatures dipped to -3C.
However, secondary roads remain very icy in places, especially on higher ground.
Gardaí at Bandon say the road from Innishannon to Bandon is treacherous and have warned anyone travelling to be very careful.
The Defence Forces are again assisting with the transportation of healthcare workers.
Nursing staff are being transported to medical facilities in the Cork area, while troops are assisting with gritting and ice clearance in Cork city centre.
In the Midlands, healthcare workers and patients are being transported in Longford, Athlone, Moate, Mullingar and Castlepollard.
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