Rival candidates launched their campaigns on Thursday for elections likely to produce Mexico's first woman president, a watershed for a nation with a long tradition of macho culture.

Opposition hopeful Xochitl Galvez took to the streets after the stroke of midnight in one of Mexico's most dangerous states, seeking to tap into voter concerns about the country's rampant violence.

Public opinion polls suggest that she faces a tough battle against ruling party candidate Claudia Sheinbaum, a former Mexico City mayor and close ally of President Andres Manuel Lopez Obrador.

Sheinbaum addressed large crowds of supporters in Mexico City's main square, vowing to continue the outgoing left-wing populist's "transformation" of the Latin American nation, including measures to fight corruption and help the poor.

On foreign policy, she said that Mexico would maintain its close cooperation with the United States, but would also demand respect for its sovereignty and "never bow our heads."

According to an average of polls compiled by the Oraculus research firm, revealed that with three months to go before the June 2 vote, Sheinbaum, enjoyed a significant lead with 63 percent support.

Polls showed Galvez has 31 percent support, while Jorge Alvarez, of the Citizens' Movement party is a distant third with just five percent,

At stake is the future of Latin America's second-largest economy, a country of 126 million people that is a key trading partner of the United States and a major tourist destination, but which faces huge challenges from illegal migration and drug-related violence.