The Taoiseach has described today as "an historic day for Ireland" where "a quiet revolution has taken place".
In a speech following the official declaration at Dublin Castle he said today Ireland "voted for the next generation".
"We have voted to look reality in the eye and we did not blink".
"We have voted to provide compassion where there was once a cold shoulder, and to offer medical care where once we turned a blind eye".
Addressing those who voted no, Mr Varadkar said: "You may feel the country has taken the wrong turn, and is no longer a country you recognise".
"I would like to reassure you that Ireland is still the same country today as it was before, just a little more tolerant, open and respectful".
He said today would be remembered as a day when "we embraced our responsibilities as citizens and as a country".
Mr Varadkar claimed Ireland was united, with men and women, nearly every age group and every social class opting for reform in yesterday's referendum.
The Taoiseach campaigned for the change, which he said represented the compassionate choice for women forced to travel to England for terminations or taking unregulated abortion pills.
Taoiseach Leo Varadkar has said today is a historic day for Ireland, and that he believes the country has voted for the next generation pic.twitter.com/Cc8qTHawbO— RTÉ News (@rtenews) May 26, 2018
He said: "What we see is the culmination of a quiet revolution that has been taking place in Ireland over the last couple of decades".
Speaking earlier today, he said that the plan was to get the legislation enacted by the end of the year, saying the fact the result was clear made this easier.
Out of all the political leaders in the Dáil, Fianna Fáil leader Micheál Martin faced into yesterday's vote with most to lose given that most of his parliamentary party were backing a No vote.
Speaking to RTÉ News today, he said that the Irish people had made the right decision and it would mean better care for women in Irish hospitals.
"It's the dawn of a new era," Mr Martin said.
He said he wrestled with the topic, but that women who told their stories had a big impact on him, especially those who had suffered rape, incest and fatal foetal abnormalities.
Sinn Féin president Mary Lou McDonald said it was an important message of compassion and progress and the vote was "cathartic".
She added: "We have without doubt done right by Irish women for this generation and many to come."
She said there was no reason to delay legislation.
Tánaiste Simon Coveney described it as a vote to put a "more compassionate and appropriate" policy in place, one that helped and respected women in vulnerable circumstances.
"This is a huge step forward for Ireland," Mr Coveney said.
Minister for Health Simon Harris said he could not have predicted the scale of the victory.
"Under the Eighth Amendment, the only thing we could say to women was take a flight or take a boat and now the country is saying no, take our hand, we want to support you," Mr Harris said.
"Women in crisis pregnancy, we were telling them or letting them feel like they were on their own - and today the country is saying no, we want to stand with you."
Children's Minister Katherine Zappone said she felt emotional and expressed deep gratitude to voters. She pledged the legislation would be introduced quickly, before the end of the year.
Senator Catherine Noone, who chaired a parliamentary committee that recommended the abortion changes, said she was very emotional when she heard the predicted results.
Ms Noone said the resounding majority had surpassed her expectations.
She added it showed that many politicians were "somewhat out of touch" with what the people think on the ground.
Labour Party leader Brendan Howlin said Irish women had been denied security and protection at crucial points in their lives over the last 35 years.
"We've denied them dignity too," Mr Howlin said. "Today is a levelling day for them. And it is very much a cause for celebration."
Mr Howlin added the mandate from the people was unambiguous and that it was up to politicians now to ensure the proposed legislation was passed.
Pro-choice campaigners hailed the Yes vote as a resounding roar for "dignity and decency".
Together for Yes co-director Orla O'Connor said the exit poll results showed "a resounding roar from the Irish people" for repealing the Eighth Amendment.
"Our campaign, and we will be forever indebted to those women and couples whose own bravery and dignity have moved hearts and changed minds - and given the scale of the victory, changed the country," Ms O'Connor said.
Co-director Ailbhe Smyth said: "This will be a moment of profound change in Ireland's social history, a moment when the nation collectively stood up for women and for their healthcare, and voted for constitutional change.
"Together For Yes always knew that Ireland was ready for this change, because of the evidence and facts showing the harm and the pain of the Eighth Amendment."
Amnesty Ireland executive director Colm O'Gorman said the outcome would be a huge milestone for women's rights.
Mr O'Gorman said, by voting resoundingly to end the constitutional ban on abortion, Ireland was sending a powerful message to women and girls in Ireland and across the globe.
"This is such an important vote for women's dignity and bodily autonomy," Mr O'Gorman said.
Vocal Yes campaigner Dr Peter Boylan described the result as a "watershed moment".
The Institute of Obstetricians and Gynaecologists chairman said he was "very relieved" and felt "vindicated".
"It's a wonderful day for Irish women," Dr Boylan said.
He paid tribute to the women who "told their stories so bravely".
Dr Boylan also thanked his the consultants at the coalface, who he said witnessed at first hand the reality the Eighth Amendment made.
Master of the National Maternity Hospital Dr Rhona Mahony said today's result was "all about the women, who have endured so much, the women who have travelled in really, really difficult circumstances and the women who have spoken out and told their stories".
Dr Mahony said for them it must be extraordinary to see the whole country get behind them and acknowledge this and say "we are going to do things differently".
She said the message to women is about care, real life and understanding the dilemmas that women face every single day.