Prime Minister David Cameron has congratulated Arlene Foster on becoming Northern Ireland's first female First Minister.
The Fermanagh MLA is also the youngest person to take up the position.
On Monday afternoon, David Cameron tweeted his congratulations to Mrs Foster.
"I look forward to working with her on a positive future for NI," he said.
PM: Congratulations to Arlene Foster, NI's 1st female First Minister. I look forward to working with her on a positive future for NI.— UK Prime Minister (@Number10gov) January 11, 2016
The 45-year-old from Co Fermanagh was formally appointed to the role on Monday afternoon at the Assembly's first plenary session of the year at Parliament Buildings, Belfast.
She has already replaced Peter Robinson as leader of the DUP who nominated her for the position.
The married mother-of-three, who defected to the DUP from the Ulster Unionists in 2004, has assumed office alongside long-serving Sinn Fein deputy First Minister Martin McGuinness.
Mrs Foster said she was "truly humbled by the trust and confidence" that had been placed on her.
"I want Northern Ireland to be a beacon to the world of how, by working together with political opponents and old enemies, we can create a Northern Ireland that we can all be proud of."
Mrs Foster said she took great pride in being the first woman to hold the office, and also its youngest incumbent.
"I can think of no greater honour than to have the opportunity to serve my country and the people of Northern Ireland as their First Minister," she said.
Martin McGuinness opened by acknowledging the "hurt" that Mrs Foster's family had suffered during the Troubles.
He stressed that his republicanism and Mrs Foster's unionism would not be diluted by the two holding Northern Ireland's top two political posts.
He said: "It is important that we do acknowledge that hurt because people on this side of the house have also been hurt, losing neighbours and friends.
"We have huge challenges in the time ahead."
This morning presented outgoing FM Peter Robinson with Pádraig ÓTuama's wonderful hand written poem 'Shaking Hands.' pic.twitter.com/dN5nEOlLA3— Martin McGuinness (@M_McGuinness_SF) January 11, 2016
Mr Robinson, 67, announced his intention to retire from front-line politics last November, days after signing a political deal with Sinn Fein and the British and Irish Governments to stabilise the rocking coalition administration.
Making his final speech as First Minister, Peter Robinson said it had been a great privilege to serve the people of Northern Ireland for almost 40 years.
He said: "Real progress has been made, the foundations have been laid and it is for others to continue to build."
He thanked Martin McGuinness and described Arlene Foster as a "worthy successor" vowing not to interfere in her leadership.
Paying tribute, Arlene Foster described Peter Robinson as an "astute leader leaving large political shoes to fill".
She added: "In his time he has helped redraw the unionist political map, few have endured more difficult political times and come out successfully.
"Few will ever know his life time of service.
"It is a daunting task for me to follow."
"When his time came to serve his country and his community he was not left wanting and left Northern Ireland in a better place."
Martin McGuinness also paid tribute.
"We faced many challenges and great difficulties and came through them," he said.
"I had a friendship with Ian Paisley that lasted until the day he died and I have a friendship with Peter that will last until the day we both die."
SDLP leader Colum Eastwood, who was appointed late last year, described Mr Robinson as a "formidable political character" whose efforts helped cement the devolved institutions at Stormont.
Ulster Unionist Party leader Mike Nesbitt joked that the three-minutes of allocated speaking time would not do justice to the number of disputes between his party and the DUP.
However he added: "I wish Mr Robinson and his family a healthy and prosperous future."
Alliance leader David Ford said the former first minister had a long and significant career spanning some of the biggest events in the history of the region.
"He has played a significant part in ensuring a better future for all of us," Mr Ford said.
Although he could not endorse Mr Robinson's legacy, Jim Allister, of the TUV, wished him well in retirement.
He said: "In the early days our paths and policies coincided significantly but in the latter years they have diverged emphatically.
"But I think that the retiring first minister and I would probably form a different perspective; each of the view that each of us took the wrong road and in holding that view of each other that is something about which we still agree.
"No one would expect me to endorse the legacy or embrace the legacy of the retiring first minister."
Writing exclusively for the Belfast Telegraph, incoming First Minister Arlene Foster said she wanted Northern Ireland to be a better place for our children compared to what she experienced.
I was born as the third of four children in July 1970. We didn't have much, but wanted for nothing. My father was a full-time RUC officer. Mum was a housewife and full-time mother, hailing originally from Sandy Row.
Dad and mum worked hard on our small farm near the border. They kept some cattle and some poultry, but it was certainly not a big money spinner!
Whilst Daddy survived the PIRA's attempt on his life, sadly he died before seeing his wee girl become the First Minister of Northern Ireland. He was always my most loyal supporter and would have been so proud. My mum, now in her 80s, will travel up to sit in the public gallery today.
When I was growing up in the '70s' and '80s, 2016 was often talked about in our home. People thought there would be a united Ireland by then. There's not and as a committed unionist, I'm glad support for the Union is growing in every area of Northern Ireland.
Support for the Union is growing because of the decisions made by my predecessors, both Dr Paisley and Peter Robinson.
They had the vision to see that Northern Ireland's place in the Union could best be secured by making Northern Ireland successful, with its own devolved administration.
As I consider sitting behind the First Minister's desk later today, I am humbled by the confidence placed in me by party colleagues and filled with an enormous sense of responsibility. I think of the stature of those who have sat in that seat before me. They were giants in unionism all through my life.
At 45, I am the youngest to hold this position. Of course, I have my own plans and I will bring my own style to the role.
For my part, I want to make sure that what is possible for me is possible for any young boy or girl growing up in Northern Ireland. For so many reasons, this is an historic moment and I take great pride in the fact that since Northern Ireland was created almost a century ago, I am the first woman to hold such a post.
However, for me, being First Minister is not about holding the Office. It's about leading this country in the right direction and ensuring our children grow up in a better Northern Ireland than I did. The real measure of success is not in obtaining the Office, but in how it enables me to help others realise their dreams, ambitions and aspirations.
I know there's an enormous job of work. The Executive and the Assembly are far from perfect. But it's better than direct rule ministers who have no stake in Northern Ireland. Indeed, as the Fresh Start Agreement demonstrated, brick by brick we are restructuring. I will continue that agenda until we have a normal and efficient system of government.
Today, I think of those who are no longer with us. I think of friends and neighbours who were killed in the defence of democracy. I also think of all of those who served the community in the security forces during the dark days of the Troubles and those whose lives were cut short.
I make this promise: In all I do I will honour their memory. My determination to make Northern Ireland work is all the greater for the sacrifice that they have made.
I remember when the bomb went off on my school bus like it was yesterday. I even remember the smell and the deathly silence in the immediate aftermath, before the screaming started. Of course, we are all shaped by our experiences. Some of us live with scars. But whilst those scars show where we have been, we can't allow them to cloud our vision in making progress. That is how we show that terrorism did not succeed.
In my role as the Economy Minister, for seven years I travelled the world seeking to bring jobs and investment home. In that time, I am proud that we created more jobs from international investment than at any time in our history. One thing made that easy. It was the quality of our young people.
When I travel across Northern Ireland I see people with abundant gifts and talents. Sometimes those people are held back by nothing but a lack of confidence and a poverty of ambition. The only thing they lack is belief.
I want to use the First Minister's Office to restore that belief and to give new hope.
I want to instil a new confidence in our people and a pride in where we live.
I want everyone to love this country with the same passion that I do.
I am grateful for the opportunity to make this statement. It is typical of the fairness and courtesy you have demonstrated during your time in office that you provided me with this opportunity and made all the necessary arrangements. I can assure you that I do not intend to trespass on your generosity by speaking too long.