Update 7.34pm: Theresa May has not clarified how the UK will honour its "special responsibility" for the consequences of Brexit for Ireland, EU chief negotiator Michel Barnier said.
Sinn Féin president Gerry Adams said Mrs May delivered nothing but the same old story, while the leader of the DUP Arlene Foster claimed she had outlined a positive vision for the future during today's keynote speech in Florence.
Mr Barnier said: "Today's speech does not clarify how the UK intends to honour its special responsibility for the consequences of its withdrawal for Ireland.
"Our objective is to preserve the Good Friday Agreement in all its dimensions, as well as the integrity of the Single Market and the Customs Union."
After Brexit the UK's only land border with an EU state will be that on the island of Ireland and it is one of the first issues up for negotiation between the UK and Europe before they can move on to talks about future trade relations.
The British Prime Minister said: "The UK government, the Irish government and the EU as a whole have been clear that through the process of our withdrawal we will protect progress made in Northern Ireland over recent years - and the lives and livelihoods that depend on this progress.
"As part of this, we and the EU have committed to protecting the Belfast Agreement and the Common Travel Area and, looking ahead, we have both stated explicitly that we will not accept any physical infrastructure at the border.
"We owe it to the people of Northern Ireland - and indeed to everyone on the island of Ireland - to see through these commitments."
She proposed a two-year transition period for the UK after its leaves the EU, suggesting that Britain would continue to "honour its commitments" under the bloc's current budget.
The DUP leader said: "This was a positive vision outlined by the Prime Minister.
"A strong signal has been sent in her speech, that although we are leaving the EU we still want to have very positive relationships with our nearest neighbours. This is particularly important to Northern Ireland.
"The reassurances to EU citizens in the United Kingdom will be warmly welcomed, as will the proposed new economic partnership."
Gerry Adams, said her speech failed to deliver or impress.
The Sinn Féin President said: "It was nothing but the same old story. There was nothing new on the Good Friday Agreement, Irish border or EU citizens' rights.
"On the basis of today's speech and in the absence of any substantial progress toward a resolution of these three key issues, the Irish government should oppose any move toward the next phase of negotiations on trade at the October EU summit."
Update 6.58pm: The Minister for Foreign Affairs, Simon Coveney, has welcomed the "clarity" provided by British Prime Minister Theresa May's Brexit speech in Florence.
However, Mr Coveney admitted that "there are still many outstanding issues" and still a lot of work to do before the EU can begin talks with the UK in the next phase of withdrawal.
Speaking from New York, the Minister said: "I welcome the additional clarity provided by Theresa May in her speech in Florence this afternoon. The speech is a positive contribution towards making progress on phase one issues - citizens’ rights, the financial settlement and Irish issues.
"This is needed to enable the UK and the EU to move forward to the important next phase, but it is clear that there are still many outstanding issues and a lot of work is still required before European leaders can make a decision that parallel discussions on the EU’s future relationship with the UK can begin."
"Ireland is fully behind the EU’s Chief Negotiator Michel Barnier and the common mandate that he has from the 27 EU Member States."
He said Ireland and the EU's "27 partners" will examine the detail of the speech.
He said: "Ultimately Ireland wants: the gains of the peace process protected, including avoiding a hard border; an orderly UK withdrawal; a sufficiently long and non-disruptive transition arrangement; and the closest possible EU-UK future relationship, including in trade, which minimises to the greatest extent possible the impact on the Irish economy."
Earlier: Taoiseach Leo Varadkar has given a "cautious welcome" to UK Prime Minister Theresa May's keynote Brexit speech today.
Mr Varadkar said he was glad Mrs May referenced issues that affect the whole island of Ireland.
"I think I'd give the speech a cautious welcome," he said. "I think it is a genuine effort by the prime minister to move things along. [I am] particularly happy that once agains she referenced issues that are of real importance to Ireland such as the Good Friday Agreement, the peace process, making sure there is no physical infrastructure between Northern Ireland and Ireland."
However, Mr Varadkar expressed the need for further clarity regarding a possible trasition period, which he plans to discuss with her when the two leaders meet on Monday.
"But we will, of course, need further clarity and further understanding as to how a transition period might work. Requesting a transition period, I think, is also a step in the right direction.
"I'll have a chance to meet with her on Monday, talk about it a bit more then, and also consult with other European prime ministers and presidents later in the month."
Business leaders in Ireland have also welcomed the clarity provides in Mrs May's speech.
"The British Irish Chamber of Commerce welcomes the further clarity the Prime Minister has provided on her Brexit intentions," said director general of the British Irish Chamber, John McGrane.
"Businesses across the islands will especially welcome the news that the UK is to seek an implementation period during which it will adhere to current EU rules. This is something the Chamber has long advocated and is critical if we are to avoid a disruption to UK-Ireland trade.
"Recognition of the need for an international dispute resolution mechanism and of the UK’s financial obligations to the EU is also to be welcomed. We hope that this pragmatism will be taken into the negotiations in the hope of achieving the progress that is necessary on the Article 50 process so that discussions can start on the future framework.
"While the UK’s aspiration to develop a close partnership with the EU post-Brexit is very positive, more detail is needed on how this will be achieved. A lot of hard work is going to be needed on all sides if we are to achieve the partnership vision that the Prime Minister has outlined. This is especially true if we are to have a frictionless border on the island Ireland.
"Maintaining regulatory equivalence will help in some degree to achieving this and it is still the Chamber’s belief that the best way of accomplishing this will be through a tailored customs arrangement between the EU and the UK."