May, dressed in red, was greeted in person by Trump when she arrived at the White House and the two leaders then met at the Oval Office.
At a joint press conference, Trump reiterated his support for Britain's decision to leave the European Union, saying, "A free and independent Britain is a blessing to the world and our relationship has never been stronger."
May said Trump had accepted an invitation to make a state visit to Britain later this year.
The US-UK defence relationship is the deepest of any two countries, May said during the press conference.
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She said a trade deal between the two countries is in the national interests of both.
Trump threw his weight behind Britain's exit from the European Union, saying it would be a "wonderful thing" for the country.
"You're going to be able to make free trade deals without having somebody watching you and what you are doing," he said.
During the press meet, Trump also spoke on controversial topics like the use of torture as he reiterated his support for harsh interrogation techniques but said he will defer to his Pentagon chief, who opposes their use.
The meeting was being watched with global interest as the two countries seek to find common ground on trade and lay the groundwork for a new deal following last year's Brexit vote in favour of the UK to leave the European Union.
"In the end it will be tremendous asset not a liability," said the US President.
May said she is convinced that a trade deal between the US and the UK will cement the crucial relationship that exists between them, particularly as the UK leaves the European Union and reaches out to the world.
"We are discussing how we can establish a trade negotiation agreement, take forward immediate high-level talks, lay the groundwork for UK-US trade agreement and identify the practical steps we can take now in order to enable companies in both countries to trade and do business with one another more easily," May said.
sanctions on Russia.
Asked by reporters whether he was planning to lift Russia sanctions, he said: "It's very early to be talking about that."
May said Britain believes sanctions should continue until Russia upholds its commitments of the international Minsk Agreement.
May also said that Trump vowed '100 per cent' support for NATO.
Ahead of the meeting with Trump, May, while speaking at the Republican Retreat in Philadelphia, said with the emergence of non-state actors, it is time that the countries like the US and Britain assert their leadership role.
May also paid her respects to the military dead of the US at Arlington National Cemetery outside Washington DC.
The UK Prime Minister laid a wreath at the Tomb of the Unknowns at the Virginia military cemetery, which holds the remains of unidentified US troops from the First and Second World Wars, as well as the Korean war.
Dressed in black, May was greeted by troops representing all military units based in Washington, led by Major General Bradley Becker, commander of Joint Force Headquarters for the national capital region.
The new US president and the premier, who took office in July, both have strong political incentives to make the visit -- likely to be heavier on symbolism and aspiration than deliverables -- a roaring success.