Left-winger Andrés Manuel López Obrador has a clear lead in the race to be Mexico’s next president, exit polls say.
The former mayor of Mexico City, who was the front-runner ahead of the vote, is estimated to have polled at least 53%, one survey by Parametria said.
All exit polls put him well ahead of his nearest rivals.
Ruling party candidate José Antonio Meade said he accepted Mr López Obrador had won and wished him well.
His Institutional Revolutionary Party (PRI) has dominated Mexican politics for much of the past century.
The candidate for the conservative National Action Party (PAN), Ricardo Anaya, has also conceded and said he had called Mr López Obrador to congratulate him.
The first official results are expected at midnight (04:00 GMT).
Sunday’s election followed one of Mexico’s deadliest campaigns in decades with more than 130 political candidates and party workers killed.
Mexico’s political map redrawn
Will Grant, BBC News, Mexico City
Delivering early and clear acceptances of their defeats, the two main challengers have recognised Andrés Manuel López Obrador as the winner of the presidential race.
On a tough night for the ruling PRI party, their candidate, the former economy minister Jose Antonio Meade, conceded first after polls closed. Next, and even more significantly, the other main rival, Ricardo Anaya of the PAN party, also recognised Mr López Obrador’s victory.
The official results must still be published for the margin of the win to be confirmed but this is a night that Mr López Obrador and his supporters have been waiting for since 2006 – the first time he stood for the presidency in Mexico. Now, on the third attempt, he has finally taken it and, in so doing, redrawn the political map of the country.
What is at stake?
Mr López Obrador – who is often referred to by his initials, Amlo – has been runner-up in Mexico’s two previous elections but now seems likely to end the dominance of the PRI and the PAN.
He has referred to them as being part of the same “mafia of power”.
Many Mexicans have grown disillusioned with incumbent President Enrique Peña Nieto’s PRI administration, particularly over the sluggish economy and endemic corruption.
Mr López Obrador, 64, put tackling corruption at the centre of his election campaign, promising to improve wages and pensions by stamping out rampant abuse.
His opponents tried to paint him as a populist who cannot be trusted with the economy.
How did election day unfold?
Long queues of voters formed outside schools and community centres as polling day progressed but two more killings of party members were reported – one a member of the Workers’ Party in the western state of Michoacan and the other a member of the governing PRI party in the central state of Puebla.
About 88 million people were eligible to vote and could choose not only a president but also 128 senators and 500 deputies in Congress as well as state and local officials. Official turnout figures have not yet been released.
A huge crowd gathered outside the polling station in Mexico City’s Tlalpan district to watch Mr López Obrador cast his vote.
He called the election “historic” and told reporters: “We represent the possibility of real change”.
How will US-Mexico relations be affected?
Mr López Obrador has been the candidate most critical of Donald Trump and has said he will make the US president “see reason”.
Mr Trump has continuously attacked Mexico over trade and migration, saying he will renegotiate the North American Free Trade Agreement (Nafta) and build a wall along the US-Mexico border.
His hardline stance on migration – particularly the separation of migrant families at the US border – has brought widespread condemnation.
Some 2,000 children remain separated from their migrant parents, despite Mr Trump agreeing to curtail the policy.
Mexico election: López Obrador set to win as rivals concede