Mexico's old ruling party, the PRI, is set to return to power as early official results indicate its candidate Enrique Pena Nieto has won the presidential election.
Mr Pena Nieto, 45, is on about 37%, several points ahead of Andres Manuel Lopez Obrador, who has not conceded.
Thousands of police were on duty for the vote, amid fears of intimidation from drug gangs.
Mexicans were also electing a new congress and some state governors.
Celebrations at the headquarters of the PRI (Institutional Revolutionary Party) started after the polls closed.
Mr Pena Nieto declared: "We all won in this election. Mexico won."
"This is just the start of the work we have before us."
He thanked Mexican voters for giving the PRI a second chance, saying his administration would have a "new way of governing".
The election campaign was dominated by the economy and the war on drugs.
"There will be no pact nor truce with organised crime," Mr Pena Nieto said.
He had been presented as the new face of the PRI, a break with the party's long and at times murky past that included links with drug gangs.
The party held on to power for 71 years until it was defeated in 2000.
Mr Pena Nieto built his reputation on the "pledges" he set out for his governorship in Mexico state, focusing on public works and improvement of infrastructure.
Outgoing President Felipe Calderon has congratulated Mr Pena Nieto and promised to work with him during the transition to his inauguration in December.
"I sincerely hope for the smooth running of the next government for the benefit of all Mexicans," Mr Calderon said, in a televised address.
Andres Manuel Lopez Obrador, running for the leftist Party of the Democratic Revolution (PRD) is in second place with about 33% of the vote.
The official quick count, published by the electoral authorities (IFE), is based on returns from a sample of around 7,500 polling stations across Mexico.
Mr Lopez Obrador, who was the runner-up in the 2006 election, has not conceded victory.
"The last word hasn't been spoken yet," he said.
"We simply do not have all the facts. We are lacking the legality of the electoral process."
In 2006, he refused to recognise Mr Calderon's victory and led street protests for months afterwards.
Josefina Vazquez Mota, the candidate of the governing National Action Party (PAN) had already accepted defeat.
The initial results from IFE put her on some 25%.
Almost 80 million people were eligible to cast their ballots on Sunday.
Police and army were deployed to protect voters from intimidation by drug cartels at polling booths.
Officials said the voting was largely peaceful, but reported some initial problems as a number of stations opened later than planned.
"Everything has been very good," one voter in Mexico City told the BBC. "But people aren't very motivated to vote, perhaps because the candidates make so many promises but we're always worse off."
With nearly half the Mexican population living in poverty, the economy was one of the main issues in the campaign.
Unemployment remains low at roughly 4.5%, but a huge divide remains between the rich and the poor.
Another issue dominating the campaign was the war on drugs, launched nearly six years ago by President Calderon, who is constitutionally barred from seeking re-election.
The main opposition candidates have been critical of Mr Calderon's policies.
They point out that more than 55,000 people have been killed in drug-related violence since 2006.
Mexicans were also electing 500 deputies, 128 senators, six state governors, the head of government in the Federal District (which includes Mexico City) and local governments.