ESB Networks has said more than 67,000 homes, businesses and farms are still without power as a result of Storm Ali.
The worst affected areas are Cavan, Sligo, Leitrim, Donegal, Castlebar, Galway, Portlaoise, Dundalk, Mullingar and parts of north Co Dublin.
At one point at the height of the storm more than 180,000 customers were without power.
ESB Networks said it could take up to three days to restore power to all affected customers.
The damage is mainly due to fallen trees on overhead lines as a result of the high winds.
ESB Networks has urged the public not to approach fallen power lines and to report any damage to the electricity network by calling 1850 372 999.
The storm has brought strong winds along Atlantic coastal counties, as well as further inland.
Gusts of up to 143km/h hit Mace Head in Galway, while south to southwest winds hit speeds of over 80km/h.
At one stage, more than 250,000 homes and businesses were without power on both sides of the border as power lines toppled.
ESB crews are en route to sites where faults have occurred but will work on repairs when it is safe to do so.
Telecoms provider Eir says 86 of its mobile phone mast sites are without power and as a result service in those areas is affected. This represents 4% of its total number of masts.
Dublin Airport has said 42 flights have been cancelled and eight flights have been diverted.
Passengers are advised to check with their airline operator before travelling to the airport.
Irish Ferries has also cancelled a number of sailings to and from Dublin and Holyhead due to adverse weather conditions.
- Woman dies after caravan blows off cliff in Co Galway
- Opening of ploughing site delayed
- Live updates of Storm Ali
- In pictures: Storm Ali
Met Éireann had earlier issued a Status Orange warning, the second highest, for counties Cavan, Monaghan, Donegal, Dublin, Kildare, Louth, Meath, Galway, Leitrim, Mayo, Sligo, Clare and Kerry.
The Road Safety Authority has asked road users to check local weather and traffic conditions and be aware of the conditions before setting out today.
Bus Éireann has said all Galway city services will not operate until further notice due to the bad weather, but all other services are operating.
Iarnród Éireann is warning customers to expect delays due to storm related works on a number of lines.
Luas services on the Green Line will not run between the Beechwood and Dawson stops tomorrow due to storm damage.
Operators of the Luas service have said that tickets will be valid on Dublin Bus for the duration of the service disruption.
Earlier today, a tree made contact with the overhead power line, which wrapped around the pantograph on a tram near the Charlemount stop.
The pantograph is the link on the roof of a carriage that connects it to the overhead power line.
An Post has said the high winds and damage have resulted in disruptions to service in over 60 post offices.
Mail deliveries across a wide area of the north and northwest were temporarily disrupted during the height of the storm.
The second day of the National Ploughing Championships in Co Offaly was cancelled as a result of the weather.
The event will now continue into Friday and wristbands for today will be valid tomorrow and Friday.
The Office of Public Works closed some of its public sites in areas where the wind warnings were in place.
The sites closed include Sceilig Mhichíl, the Battle of the Boyne site in Co Meath and St Audeon's Church in Dublin.