An amber wind warning, valid from 19:30 GMT on Tuesday, is in place for much of Northern Ireland.
The Met Office has warned that peak gusts of 80-90mph are possible, especially along exposed parts of the east coast as Storm Eleanor passes.
Flying debris could lead to injuries or danger to life, while some damage to buildings is likely, it has said.
Some disruption to road, rail and air travel is likely and ferry services could be affected.
There is also a good chance that power cuts may occur with mobile phone coverage possibly affected.
In a tweet posted at 20:00 GMT, the PSNI said: “Winds are beginning to pick up. Please do not travel unless it is absolutely necessary. If you have to travel drive carefully and be aware of road conditions.”
A number of trees are down on roads in the Armagh, Banbridge and Craigavon areas, one of which is blocking a lane of the A1 near the Outlet shopping centre.
An electric pole has been brought down at Annagh Bridge in Portadown.
There are reports of up to 15 trees down on the road network in County Fermanagh, according to Trafficwatch NI.
All three Danske Bank Irish Premiership football matches being played on Tuesday night have now been abandoned.
Storm Eleanor is the fifth named storm to hit the UK this winter.
The Met Office warning lasts until 04:00 GMT on Wednesday. It originally issued a yellow warning, but has now upgraded it to amber.
In the Republic of Ireland, Met Éireann has issued two Status Orange warnings and road users are being urged to travel with care as Storm Eleanor sweeps across the country, with strong winds and flooding in some areas.
Motorists have been warned to avoid Galway city centre due to serious flooding, while flooding has also been reported in parts of counties Cork and Kerry.
Thousands of homes are without power in parts of counties Mayo and Donegal.