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Queen Elizabeth II extended her "deep sympathy" to all those who have suffered in the turbulent intertwined history of Britain and Ireland, in the keynote speech of her historic state visit.

The first British monarch to visit the Republic since it won its independence in 1922, said the two countries were now "firm friends and equal partners" during a state banquet at Dublin Castle, the former seat of British colonial power.

"It is a sad and regrettable reality that through history, our islands have experienced more than their fair share of heartache, turbulence and loss," the 85-year-old sovereign said.

"These events have touched us all, many of us personally, and are a painful legacy. We can never forget those who have died or been injured and their families.

"To all those who have suffered as a consequence of our troubled past, I extend my sincere thoughts and deep sympathy.

"With the benefit of historical hindsight, we can all see things which we would wish had been done differently or not at all."

Earlier, the Queen undertook one of the most daring diplomatic engagements of her reign when she stepped out into Ireland's Croke Park stadium, the scene of a massacre by British troops in 1920.

In a gesture that summed up how far relations between the two old enemies have come, the Queen was brought into Croke Park through the Hogan Stand, named after a player killed during the Bloody Sunday massacre nearly a century ago.

She met players, chatted about Irish sport and was entertained by a marching band and traditional dancing, although the seats around the vast stadium were empty - a reflection of the tight security around the trip.

So far on her four-day state visit, the Queen has won praise for showing a determination to address the bloody past and offer powerful gestures of reconciliation.

"It'll put some demons to rest, bring a bit of closure," said Phil Dolwer, 32, a chef working in a cafe around the corner from Croke Park, ahead of the Queen's appearance there. "The time is right."

Croke Park, the home of Irish sports, is an iconic place for nationalists. In 1920, during Ireland's war for independence, British troops opened fire on a crowd there after 14 British intelligence officers were killed in the city the night before.

Fourteen civilians were killed, and Bloody Sunday was born - a rallying cry for the nationalist cause.

The threat from militant nationalists opposed to the Northern Ireland peace process means security around the Queen is tight. There are no public walkabouts and streets around venues have been cleared of onlookers.

Several hundred people took part in demonstrations on Tuesday that were condemned by many Irish people. Around 30 protesters gathered outside Croke Park to protest on Wednesday.

"What is she doing here, at the scene of a massacre by British forces?" said Jason Flood, an unemployed 35-year-old GAA fan.

"I don't know what they are trying to achieve. I can't understand their mentality."