The Queen has laid a wreath at the Republic of Ireland's Garden of Remembrance during the first visit by a British monarch since Ireland gained independence.
The garden, in Dublin, is dedicated to people who fought for Irish independence from Britain.
Dozens of protesters gathered nearby amid one of the country's biggest security operations.
Earlier, the Queen was welcomed by President Mary McAleese.
It emerged that a pipe bomb found on a bus bound for Dublin on Monday had been made safe by the Irish army.
Upon landing at Baldonnel military airbase, just outside Dublin, the Queen emerged wearing an outfit of symbolic emerald green.
After lunch the monarch wore a white outfit.
God Save The Queen was played when she arrived at the Garden of Remembrance to lay the wreath - regarded as a highly symbolic act as the garden commemorates Irish people who fought over the centuries against British rule.
A one-minute silence followed the laying of the wreath, after which the Irish Tricolour was raised from half to full mast and the country's national anthem was played.
Much of the centre of Dublin was cordoned off, leaving many streets empty, as part of the strict security clampdown put in place for the visit.
Meanwhile, riot police officers jostled with demonstrators at two separate protests on streets several hundred yards from the garden.
The sounds of protesters could be heard during the laying of the wreath and black balloons were released by some demonstrators.
There were scuffles between police and republicans protesting against the visit in the centre of Dublin at Tuesday lunchtime.
Up to 200 supporters of the Eirigi socialist republican party later retreated to stage a rally nearby.
The Queen then went on to Trinity College to be shown the Book of Kells, an ancient illuminated manuscript in Latin, containing the four gospels of the New Testament.
Later, the monarch met Trinity scholars, musicians and artists.
At Baldonnel airbase, the royal party was greeted by an Irish Air Corps guard of honour and presented with flowers by a south Dublin schoolgirl, eight-year-old Rachel Fox.
The Queen and Duke of Edinburgh were then driven in a bombproof and bulletproof car, with a 33-motorcycle escort, to Aras an Uachtarain - the official residence of the Irish president in Dublin's Phoenix Park. The visit is taking place following Mrs McAleese's invitation.
The Queen shook hands with the president at the front of the residence before moving inside to meet the Taoiseach (prime minister) Enda Kenny.
The Irish flag and the union jack flew side by side at the gates to the building, where both countries' national anthems were played.
A 21-gun salute and an Air Corps flypast also marked the Queen's arrival.
She and Prince Philip signed the visitors' book in the residence's lavish state ballroom, before having lunch with the president and her husband.
King George V was the last monarch to visit the country, in 1911, when what is now the Republic was then part of the UK.
UK Prime Minister Cameron will join the Queen and Duke of Edinburgh for part of their trip on Wednesday, while Foreign Secretary William Hague is following the usual practice of accompanying the royals throughout their visit.
The Queen will attend events this week at the National War Memorial Gardens in Islandbridge and Croke Park stadium.
Croke Park is the home of Gaelic games where in 1920, during the Irish War of Independence, British forces fired into the crowd at a football match, killing 14 spectators and players.
The Queen is also to make a speech at a state dinner at Dublin Castle.
There are also plans for the Queen and Prince Philip to visit the Irish National Stud in County Kildare, as well as the Rock of Cashel in County Tipperary and a technology park in Cork.
The bomb on the bus was found in a holdall in the luggage compartment on Monday night during a check in Maynooth, County Kildare, to the west of Dublin.
About 30 people who were on board the bus were taken off and transported to Dublin in another vehicle.
The device was later made safe in a controlled explosion carried out by an Irish army bomb disposal team.
A coded bomb threat relating to London had been received on Sunday, the first issued by Irish dissidents outside Northern Ireland in 10 years, officials said.
However, the threat level for Northern Ireland-related terrorism in Britain remains unchanged at substantial. In Northern Ireland it is severe.
The cost of the security operation has been estimated at 30 million euros (£26m), with measures including:
- Deployment of more than 6,000 Irish police and Defence Forces personnel onto the streets of Dublin
- Increased surveillance of known republican dissidents
- A ring of steel, comprising 25 miles of crowd-control barriers, installed around the Irish capital
- Checking of thousands of manhole covers and lamp-posts, and parking bans imposed on 30 city centre streets
Richard Boyd Barrett, who represents the People Before Profit party, criticised the expense of the state visit.
"We don't need to spend 30 million euros on a big circus bringing over one of the wealthiest women in the world to parade around the streets of Dublin and shut the city down for a few days to tell us what we already know, which is we are friends of the people of Britain," he said.
The Queen's itinerary
Tuesday 17 May: Visit to the official residence of the Irish president, Mary McAleese and tree planting ceremony. Wreath laying ceremony at the Garden of Remembrance and visit to Trinity College to view the Book of Kells.
Wednesday 18 May: Tour of the Guinness Storehouse; visit the Taoiseach, Enda Kenny, at Government Buildings, wreath laying ceremony at the Irish War Memorial Garden; Croke Park stadium to meet Gaelic Athletic Association; state dinner at Dublin Castle.
Thursday 19 May: Visit to National Stud at Kildare.
Friday 20 May: Tour of St. Patrick's Rock, Cashel and the English Market and Tyndall Institute, Cork.