The status yellow warning advises that showers of snow are expected overnight and on Tuesday morning.
Severe weather warning videos created in collaboration with RTÉ's Teresa Mannion and including advice for driving in snow and icy conditions are available on the RSA YouTube channel, RSAIreland.
Gardai and the AA are warning motorists to be extra careful on the roads tonight and tomorrow morning.
Many slips and falls happen in places people regard as safe and secure, typically outside their front door, on the doorstep, on the path or while getting out of the vehicle.
"Some risky driving conditions can be expected as road temperatures are likely to be close to freezing or below", the warning reads.
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Snow, sleet and hail are also expected to pelt parts of Britain on Tuesday, with wintry showers forecast for Scotland, Wales and western England.
Drivers in the north-westerly regions that have been told to take the most care as snow and ice could play havoc with road surfaces. It will be a bright day for many with sunny spells and showers, some wintry especially in the early morning, but turning more to rain or sleet in the middle of the day.
It will also be very cold on Tuesday.
The remainder of the week is set to be cold with some frost at night but it will turn milder by the weekend.
"Wednesday and Thursday will be cold days with moderate to fresh northerly winds, making if feel extra cold".