Storm Atiyah has made landfall, with winds hitting speeds of up to 80mph (130km/h).
Earlier on Sunday a “status red” wind warning was issued by Met Éireann for County Kerry.
Extreme caution is advised, especially in coastal areas and on high ground.
ESB Networks has said its crews have dealt with several thousand power outages across the Republic of Ireland. Irish broadcaster RTÉ reports that the south-west area is the worst affected.
The “status red” warning for Kerry was in place from 16:00 to 19:00 local time on Sunday. It is now under a “status orange” wind warning.
Kerry County Council has reported a number of incidents following the “status red” wind warning.
It said a tree fell on a car near Mountcoal Cross on the N69.
Met Éireann said there was a possibility of coastal flooding due to a combination of high seas and a storm surge.
Trains in Cork and Kerry were forced to travel at reduced speeds, resulting in delays.
Storm Atiyah was tracking between Iceland and Ireland on Sunday.
Although the UK is not expected to be as badly hit by the storm, the Met Office has issued a yellow wind warning for Wales, with gales of up to 70mph set to hit coastal areas.
The warning is in force until 19:00 GMT on Monday.
Orange wind warnings have also been issued for Donegal, Galway, Leitrim, Mayo, Sligo, Clare, Cork and Limerick, which came into effect from 13:00.
The warnings will remain in place until 06:00 on Monday, with a yellow wind warning in place for the rest of the Republic of Ireland until 13:00 on Monday.
Kerry County Council advised people to stay indoors during the status red warning.
An emergency helpline has been set up by the council to report fallen trees, flooding or debris on roads. Anyone wishing to use it should call 066 718 3588.
A status red marine warning has also been put in place, with winds reaching gale force eight to storm force 10 in all Irish coastal waters.
The Republic of Ireland’s National Parks and Wildlife Service said Killarney National Park and Gardens and Muckross Park and Gardens are closed.
Seven other parks in the west of the country are also closed while the weather warnings remain in place.
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Northern Ireland Analysis
By Geoff Maskell, BBC NI weather presenter
The UK Met Office works in partnership with both Met Éireann and KNMI (The Dutch national weather forecasting service) to name storms.
The criteria used for naming storms are based on both the impact the weather may have, and the likelihood of those impacts occurring.
A storm will be named when it has the potential to cause an amber or red warning.
When the criteria for naming a storm are met, any of the three partners – the Met Office, Met Éireann or KNMI – can do so.
That does mean that sometimes, like today, Met Éireann have named Storm Atiyah and issued a Red Warning in County Kerry.
No warnings have been issued for Northern Ireland by the Met Office, however gusts close to 60mph (100km/h) can be expected in western areas on Sunday evening.
This is the first named storm of the season, last year there were eight storms – the last was Storm Hannah in April.
Met Éireann issue weather warnings based on a criteria, for example, if winds are set to reach a certain speed, whereas the Met Office issues warning based on the impact the weather is expected to have.
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