In Dublin, a military ceremonial event was held at the War Memorial Gardens at Islandbridge.
A military service was also held at the Mall in Cork.
Irish President Michael D Higgins travelled to Thiepval in France to pay tribute to the Irishmen who died while serving in the British, Commonwealth or US armies.
He was accompanied by the the Irish minister for arts, heritage and the Gaeltacht, Heather Humphreys.
The village of Thiepval in France is close to where the 36th Ulster Division began their attack on German lines on 1 July 1916 and is now the site of the Ulster Memorial Tower, a 70ft (21m) monument which commemorates their sacrifice.
Men from all over the island of Ireland fought at the Somme, and the service in France will also be attended by representatives from the British and French governments also.
Many people from the Republic of Ireland and Northern Ireland, including relatives of Somme veterans, have travelled to Thiepval for the commemoration event.
The ceremony in Dublin began at 07:28 local time - the time the battle started 100 years ago.
It was attended by members of the Irish Defence Forces and the Royal British Legion.

During a "very busy two hour period" on Friday, the fire service said it received more than 100 calls to over 30 flooding incidents in Southend.
Six people were rescued after their vehicles got stuck at a flooded roundabout on Woodgrange Drive.
The roof of the Pearl Dragon restaurant collapsed under the weight of the rain. No-one was hurt.
Crews were also sent to the Sea Life Centre, opposite the restaurant on the Eastern Esplanade, to pump water from the building.
Martin Richardson, owner of the Happidrome on Southend's Golden Mile, said the amusement arcade had been overcome by water.
"It came through the back, the side, it came through the wall, it came through the roof, the main drains," he said. "At least four of the arcades on the promenade suffered.
"It's going to be a long while before we're open."
Mark Baumann, owner of the Bourgee restaurant on the seafront, said he watched in vain as flood water entered the building.
"I've never seen rain like it," he said. "The water started to come into the restaurant, and then it went further, and further.
"It was all hands on deck - some of the customers helped to squeegee the water away."
The rain has resulted in traffic delays, with the A12 London-bound at Witham blocked by flood water.
BBC Travel has the latest on the disruption.

Addressing the issue of students getting value for money, he will challenge an "upward spiral" in university leaders' pay.
Universities will be asked to publicly justify how they can pay their heads more than the prime minister.
The question of university value has been under scrutiny, with fees increasing to £9,250 in England.
In a speech to university leaders on Thursday, Mr Johnson will call for universities to set out much clearer contracts for students on what they can expect.
He will promise better "consumer protection" for students.
An annual study has shown a decreasing sense of value for money among students - with concerns about getting fewer teaching hours than they expected.
Mr Johnson will highlight concerns over increases in vice-chancellors' pay - with dozens of university heads now receiving over £300,000 and some being paid more than £400,000.
"When students and taxpayers invest so heavily in our higher education system, value for money should be guaranteed.  Yet, I am still hearing students say that their course is poor quality.
"This is not good enough, especially when some vice-chancellors take home a wage that in some cases exceeds that of the prime minister."
There have been concerns about the rising costs of university in England - with tuition fees increasing to £9,250 and interest rates on loans rising to 6.1%.
The Institute for Fiscal Studies says that average debts for graduates will now be over £50,000 - and that interest charges will have reached £5,800 before students have even graduated.
Former education minister Lord Adonis says such high levels of interest are "indefensible" and predicts that they will be changed.
And the head of the Russell Group of leading universities has described such interest charges as "out of touch" and called for them to be reviewed.
But Mr Johnson is expected to defend the current system of student finance and university funding.
The centenary of the Battle of the Somme has been marked with ceremonies in the Republic of Ireland.
[NEXT_CONCEPT]
Six people were rescued from cars trapped in flood water, as heavy rain hit Essex overnight.
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Universities Minister Jo Johnson is going to challenge universities over "excessive" pay for vice-chancellors.