The letter was hand-delivered to European Union president Donald Tusk in Brussels by Tim Barrow, Britain's ambassador to the EU, after being signed by May in her Downing Street office.
Here are the main points from the historic letter:
May called for continued strong ties with the EU in contrast with some of the harsh rhetoric coming from pro-Brexit campaigners.
"The government of the United Kingdom wants to agree a deep and special partnership between the UK and the EU, taking in both economic and security cooperation," she said.
She emphasised in particular the importance of security in Europe, saying it was "more fragile today than at any time since the end of the Cold War".
The future status of around three million EU citizens living in Britain and more than one million British nationals in other parts of the bloc has been a major source of discord since the referendum.
"We should remember that at the heart of our talks are the interests of all our citizens," she wrote.
"We should aim to strike an early agreement about their rights."
May has said that she can only guarantee the status of EU nationals when she receives reciprocal assurances on guarantees for British citizens.
British business leaders have warned about the prospect of Britain crashing out of the EU without an agreement in place, which would lead to higher trade tariffs for most goods.
May had earlier said that "no deal is better than a bad deal" but she appeared to tone down her approach in the letter.
"Both the United Kingdom and the European Union would of course cope... but it is not the outcome that either side should seek," she wrote.
She warned in particular of the security implications, saying that "our cooperation in the fight against crime and terrorism would be weakened".
"We must therefore work hard to avoid that outcome," she wrote.
A key priority in the letter was the border between the Republic of Ireland, an EU member state, and Northern Ireland, a British province.
"We want to avoid a return to a hard border between our two countries," she said.
She also said she wanted to ensure that Britain's withdrawal "does not harm the Republic of Ireland".
"We also have an important responsibility to make sure that nothing is done to jeopardise the peace process in Northern Ireland, and to continue to uphold the Belfast Agreement," she said.
The 1998 peace deal brought to an end three decades of conflict in Northern Ireland in which more than 3,500 people were killed.
May said she wanted Brexit negotiations to be conducted in parallel with discussions on the future relationship between Britain and the EU.
"We... believe it is necessary to agree the terms of our future partnership alongside those of our withdrawal from the European Union," she wrote.
The EU has said the two processes should come one after the other - first a deal on Brexit and then one on future ties.
The letter said May wanted Britain and the EU to "minimise disruption and give as much certainty and possible", particularly for the business community.
"In order to avoid any cliff-edge as we move from our current relationship to our future partnership, people and businesses in both the UK and the EU would benefit from implementation periods to adjust in a smooth and orderly way to new arrangements," she said.
"It would help both sides to minimise unnecessary disruption if we agree this principle early in the process."
(Except for the headline, this story has not been edited by NDTV staff and is published from a syndicated feed.)