There are real fears that Britain's exit from the European Union could break up the union.
There have already been strong calls for another Scottish referendum, and Northern Ireland's position in the union looks more precarious by the minute.
But what exactly are leaders on both sides of the border saying? We've got the latest.
Stormont's Deputy First Minister called for a border poll on a unified Ireland.
"This decision to drag us out of the European Union against our democratically expressed wishes has nothing to do with issues around the European institutions and everything to do with the civil war within the British Tory party," he said, after 11 out of Northern Ireland's 18 constituencies voted to remain.
The Sinn Fein politician added Britain needed to "allow the people of the north to have their say on their own future", and was opposed to Britain "dragging us out of Europe".
Northern Ireland Secretary of State Theresa Villiers is the one who has the power to call a border poll, which is permitted where there is clear evidence of a public opinion swing towards Irish unity.
She said she did not believe the criteria for triggering a border poll had been met.
Stormont's First Minister insisted Northern Ireland's place in the Brexiting UK was safe despite the region voting for Remain.
The Democratic Unionist leader, who campaigned for an EU exit, said she was "absolutely certain" the constitutional status was secure and claimed the Union was stronger after the Leave vote.
"The call for a border poll was as predictable as the flowers in May. We knew it would come but the test has not been met so therefore I don't believe it will happen," she said.
The Republic of Ireland's Taoiseach said he was "very sorry" that the UK had voted to leave the European Union but stressed that the "strong and close relationship" between the two countries would continue.
Keeping an open border between Ireland and Britain was a priority, he said.
"There will be no immediate change to the free flow of people, goods and services between our islands," he reassured people.