UK leader makes controversial push for Turkey trade deal
May flew overnight to Ankara by RAF Voyager jet from the US, where she and Trump hailed a new chapter in the trans-Atlantic "special relationship".
May's talks in Ankara with Erdogan and Prime Minister Binali Yildirim will focus on boosting trade between Turkey and the UK, particularly once the EU member leaves the European economic bloc.
Also among the discussions was on increasing cooperation over security and counter-terrorism.
As is traditional for any visiting leader, she started the visit by laying a wreath at the mausoleum of Turkey's modern founder Mustafa Kemal Ataturk.
May, who is paying her first visit to Turkey since becoming prime minister, is under pressure at home to condemn Turkey's clampdown on civil liberties since the government crushed a coup attempt in July.
Turkey has detained tens of thousands of people suspected of links to a movement led by US-based Muslim cleric Fethullah Gulen, whom the government accuses of orchestrating the failed attempt.
The clampdown extended to other government opponents. More than a hundred journalists and pro-Kurdish party leaders are in jail.
Kate Allen, head of Amnesty UK, said the visit was a "vital opportunity" for May to ask "probing questions" about allegations of excessive use of force and ill-treatment in detention.
May's spokeswoman said the UK had urged Turkey "to ensure that their response is proportionate, justified and in line with international human rights obligations".
She also said there weren't "any issues that the prime minister is afraid to bring up".
May and Turkish leaders are also expected to discuss the conflict in Syria and efforts to reunite Cyprus.