The status yellow alerts cover parts of the midlands, east and north-west, and are in place throughout the day.
A third yellow warning for wind in Galway, Clare and Kerry will also take effect from midday.
Met Éireann says gusts of up to 100kmph can be expected and there is also a risk that some rivers could burst their ban.
Forecaster Andrew Doran Sherlock says the poor conditions will affect many counties:
"We have got quite widespread rain over much of Leinster and Ulster at the moment, with some heavy falls.
"It is spreading south westward, so it is going to track throughout the day and it is going to continue overnight."
He said there will be some drier intervals over Leinster and Ulster tonight. However he warned that there will also be some heavy falls and large accumulations in the rain.
Wet and windy today. Showers in the southwest at first. Persistent rain in Ulster and Leinster will spread southwestwards, continuing for the day, heavy at times with spot flooding in places. Highs of 11 to 14°C with fresh to strong, gusty northwest winds and gales on some coasts pic.twitter.com/9wHfvRncND
— Met Éireann (@MetEireann) October 4, 2020
