Around 120,000 homes and businesses are now without power as Storm Ophelia hits Ireland.
ESB has warned that more outages are expected and that repair operations will take several days.
Gusts of 92mph (148kph) have already hit the south west coast and forecasters have warned that worse is yet to come.
Significant tidal surges are expected across the country after the storm hit on Monday morning and is due to sweep up through the island.
Taoiseach Leo Varadkar has urged the public to stay safe, saying: "The advice is: stay indoors until the storm passes.
"Whether that is at work, in their home or some other home, stay indoors. Check on neighbours and relatives.
"Bear in mind it is coming your way and it is a national red alert.
"It is a very dangerous storm. The last time there was a storm this severe 11 lives were lost."
Mr Varadkar added that the National Emergency Co-ordination Group will be meeting throughout the course of the day.
Met Eireann said the strongest winds have not yet hit the country.
Trees and power lines are down across Kerry and Cork as the storm makes its way across the island of Ireland.
Customers are without power in areas ranging from Co Galway, around Munster to Co Waterford.
The head of Corporate Affairs at ESB said earlier widespread power outages are expected.
In the Irish Republic, schools, nurseries and colleges have been closed, court sittings postponed, numerous hospital outpatient appointments cancelled and the Defence Forces put on standby.
Businesses have been urged to consider whether opening would pose a risk to employees.
Schools have also been advised to close in Northern Ireland while many college classes have also been scrapped. The Lord Chief Justice also advised the judiciary to cease all court hearings at 12.30pm.
Across the island, many ferries, trains, buses and flights have been cancelled.
Met Eireann has issued a status red weather warning across all of the Irish Republic.
It has described the storm as the most powerful to have ever been this far east in the Atlantic on record.
Forecasters have warned of a potential threat to life and advised the public to stay off the roads and away from the coast during the height of the storm if possible.
Counties Galway, Mayo, Sligo and Donegal are due to bear the brunt of the winds.
The Met Office has issued an amber weather warning for Northern Ireland and warned of "potential danger to life".
The differing severity of alerts north and south of the border is more due to differences in the way Met Eireann and the Met Office rate threats, rather than an indication that Northern Ireland will not be hit as hard.
In regard to Northern Ireland, the Met Office said: "There is a good chance that power cuts may occur, with the potential to affect other services, such as mobile phone coverage.
"Flying debris is likely, such as tiles blown from roofs, as well as large waves around coastal districts with beach material being thrown on to coastal roads, sea fronts and properties.
"This leads to the potential for injuries and danger to life."
The Irish Medical Organisation (IMO) has asked the public to avoid using GP services during the storm except for emergencies.
While hospital emergency departments remain open, the IMO has also urged the public to ensure that these are reserved for genuine emergencies.
Dr Padraig McGarry, chairman of the GP Committee of the IMO, said that while most GP surgeries will remain open today, they want to ensure GP members are free to deal with genuine emergency cases rather than with more routine business.
Regarding hospital emergency departments, Dr Peadar Gilligan, chairman of the Consultant Committee, said: "We expect emergency departments to be particularly busy over the coming 24 hours and we would urge the public to respect the fact that these should be reserved for genuine emergencies only."
Irish Defence Forces are on standby to deploy resources, including transport and engineering assets.
On Sunday, the Department of Education in Dublin said that all Irish schools, colleges and other education institutions are to stay closed on Monday.
The decision was taken following a special meeting of the government task force on emergency planning.
Authorities in Northern Ireland have faced criticism after announcing closure advice to schools late on Sunday night, giving parents limited time to arrange childcare.
Ophelia is on its way from the Azores in the Atlantic Ocean.
It will hit exactly 30 years after the Great Storm of 1987 killed 18 people.
ESB, the Republic of Ireland's electricity network, said the initial impact of the storm caused damage to the network across southern counties.
"As of 12 noon, approximately 120,000 electricity customers are without power. The main areas impacted are in the southern half of the country and include counties Kerry, Cork, Tipperary, Waterford, Wexford and Limerick.
"As the storm tracks across the country, ESB Networks is prioritising emergency situations. Once the electricity network is made safe, our crews will commence restoration of power to our customers.
"However, the majority of customers who have lost supply at present will be without power overnight. From previous similar storms, e.g. Storm Darwin, where over 200,000 customers were left without supply, we can predict that it will take a number of days to restore power to all customers," ESB said.
ESB advised anyone who may come across fallen wires or damaged electricity network not to touch them but report it on 1850 372 999.
The Head of the Northern Ireland Civil Service, David Sterling, held an emergency meeting with the permanent secretaries from all Government departments on Monday morning.
It was decided that all government buildings providing non-essential services close at 1pm and all non-essential civil service staff will leave work at that time.
Guidance is to be issued later in relation to school opening arrangements for Tuesday.
Belfast International Airport said the storm is having an impact, with both Ryanair and Easy Jet cancelling flights. The airport is expecting more cancellations.
Anyone due to travel has been advised to contact their airline.