Martin McGuinness has described meeting the Queen as, "very nice."
Martin McGuinness has said that, "it was nice" to meet the Queen and that the meeting went, "very well" as he left the Lyric Theatre this afternoon.
He also added, "I'm still a Republican."
Northern Ireland Secretary Owen Paterson met Mr McGuinness this morning and added that he was relaxed.
He said the Deputy First Minister greeted the Queen in Irish and their meeting was cordial.
It had obviously gone very well.This will move Northern Ireland on to a whole new plane. After all the trauma of Northern Ireland, everyone is looking forward.It is about a shared future, not a shared-out future.
None of this could have happened a few years ago so it is all looking to the future.
He said it was absolutely appropriate that when the Queen visits parts of the UK, she meets local politicians, democratically elected, pursuing their democratic political goals by peaceful means.
Mr Paterson added that it built on the success of the Queen's visit to the Republic of Ireland last year.
The Queen is expected to visit Stormont later today.
It is understood that during the VIPs' initial private meeting, Mr McGuinness welcomed both the Queen and President Higgins in Irish.
The Deputy First Minister is said to have commented briefly on the Queen's visit to Dublin last year, and in particular her comments regarding all the victims of the conflict.
A Sinn Fein spokesman said: "He emphasised the need to acknowledge the pain of all victims of the conflict and their families."
Mr McGuinness is said to have spoken to the Queen of the significance of her visit, and of the need for it to be built upon in the time ahead.
Sinn Fein said Mr McGuinness told the Queen that their meeting was a "powerful signal that peace-building requires leadership".
The Deputy First Minister also praised the role of the President in today's encounter, and welcomed that the engagement took place at an event celebrating culture across Ireland.
Ciaran Cunningham from the Republican Network for Unity has spoken to ITV News' Royal Correspondent Tim Ewart about the handshake between Martin McGuinness and the Queen.
Mr Cunningham said that, "Martin McGuinness is playing with words," when he says he is still a republican.
Ciaran Cunningham from the Republican Network for Unity has said that:
Martin McGuinness has lost all logic and he lost all sense and he has lost all respect.
Martin McGuinness is playing with words, he's trying to make the word Republican mean something that it doesn't mean.
The Queen publicly shakes the hand of former IRA member Martin McGuinness in an historic moment in Belfast.
The Queen has shaken hands in public with Sinn Fein's Martin McGuinness in a gesture of reconciliation.
Mr Guinness held the monarch's hands for a few moments, spoke to her in Irish and told her the words meant: "Goodbye and God's speed."
Asked how David Cameron viewed the handshake, the Prime Minister's official spokesman said:
Clearly, there was a visit by Her Majesty to the Republic of Ireland last year. That has taken relations between the two countries to a new level.
We think it is right that the Queen should meet representatives from all parts of the community.