NIE Networks teams have already restored electricity supplies to about 17,000 customers on Thursday morning.
It said trees and branches falling onto overhead electricity lines is the main cause of the damage.
A yellow warning of rain for all counties is in place until 14:00 GMT and a yellow warning for wind until 20:00 GMT in counties Antrim and Down.
A number of roads have been blocked by fallen trees and power outages have been reported.
Customers can call the NIE helpline on 03457 643 643 to report faults or report online at nienetworks.co.uk.
At the George Best City Airport KLM flights to and from Amsterdam have been cancelled on Thursday afternoon.
Ferry services are also expected to be disrupted later.
In the Republic of Ireland, about 56,000 customers are without power due to 770 faults, according to ESB Networks.
Large faults have been reported in counties Galway, Mayo, Sligo, Leitrim and Dublin.
Portarlington in County Laois and Celbridge in County Kildare has also experienced significant outages.
At Dublin Airport a number of flights to and from the UK have been cancelled.
Aer Lingus has cancelled almost all flights between the UK and the Republic of Ireland.
Fallen trees on roads around the Republic are also causing traffic disruption.
Met Ã‰ireann has updated its wind warning to orange - the second highest - for many counties in the east.
Winds of up to 120km/hr are forecast in counties Dublin, Kildare, Louth, Wexford, Wicklow and Meath.
An 87mph gust was recorded on the Galway coast earlier.
A yellow warning is in place for the rest of the Republic with gusts of up to 110 km/hr forecast until midday.
Some ferries between Ireland and the UK have also been cancelled, with passengers being advised to check service details with their operators.
In County Tyrone a jack knife lorry is causing delays and the A5 Tullyvar Rd is closed between Ballygawley  and Aughnacloy.
About 3,500 homes and businesses are now without power in Northern Ireland as Storm Doris lashes the UK.