ESB warned customers should make preparations for being without power for a number of days.
190,000 homes and business are without supply this morning and crews are being assisted by staff from Northern Ireland Electricity in an effort to have power restored.
However, ESB is warning it will be late into this evening before electricity is restored to most customers and a large number of homes and business will still be without power tonight.
Homes in the south and south-west of the country are worst-affected.
The company is using helicopters to patrol transmission and distribution circuits today.
"Surveys of the damage show that we have many trees down across lines, lines down, broken poles and equipment damage as result of flying debris and transformers which need replaced," the company said in a statement.
Motorists are being urged to exercise caution on the roads in case of fallen trees and other storm damage. Irish Rail reports all of its services are operating this morning except two: between Tralee and Mallow, and from Limerick to Athenry.
There may be some minor delays on other services.
Meanwhile, Gardaí are appealing to the public to "be neighbourly" in the aftermath of the storm, and to visit their elderly friends and relatives to check on their wellbeing.