GEOFF BENNETT: As many as 100 million Mexicans go to the polls Sunday to elect a new president.

And it's already guaranteed to be historic.

Both leading candidates are women.

And the country has never in its 200-year history had a female leader.

The main issues in Mexico's largest ever election are security, migration and the economy.

But, as Nick Schifrin reports, the leading candidate, Claudia Sheinbaum, represents multiple firsts.

And a note of caution: This story contains images and accounts of violence.

NICK SCHIFRIN: Never before has Mexico had a woman, the granddaughter of Jewish immigrants, and a Nobel Prize-winning climate scientist as president.

But Claudia Sheinbaum calls herself a disciplined defender of Mexico's future.

CLAUDIA SHEINBAUM, Mexican Presidential Candidate (through translator): We will have an honest government without corruption or impunity.

We will not submit to any economic or foreign power, no matter how powerful it may be.

NICK SCHIFRIN: Running a distant second, former Senator Xochitl Galvez, who leads the country's broad opposition coalition.

She's questioned whether Sheinbaum is just the protege or will be the puppet of current President Andres Manuel Lopez Obrador, known as AMLO.

CLAUDIA SHEINBAUM (through translator): The transformation initiated by President Andres Manuel Lopez Obrador is not going backwards.

LILA ABED, The Wilson Center: The big question mark surrounding a possible Sheinbaum administration is what role Andres Manuel Lopez Obrador, is going to play, if any, and whether she's going to really make a mark of her own.

NICK SCHIFRIN: Lila Abed is the acting director of the Mexican Institute at The Wilson Center.

LILA ABED: She can't completely distance herself from the political platform, from the government plan that got AMLO into power and that has sustained high levels of approval rates.

And that is what she's running on.

She is not running necessarily on a new political platform.

She's building on what AMLO has achieved.

NICK SCHIFRIN: But voters' single most important issue might be what Lopez Obrador has failed to achieve, security.

A Mexican security firm says more than 700 people connected to the election have been threatened, kidnapped or murdered; 34 candidates were killed between September and May, much of the violence unleashed by cartels that fight for lucrative trafficking routes in the most violent states, Guerrero, Puebla, and Chiapas.

In Chiapas, candidates have been running for office and from gangs.

Earlier this year, Diego Perez, the mayoral candidate in San Juan Cancuc, was found dead, dumped in a ditch with signs of torture.

And this month, 28-year-old Lucero Lopez, a mayoral candidate in La Concordia, was shot to death at a campaign rally.

LINDA HIGUERA, Green Party (through translator): It's hard.

We need to be able to walk in our streets with security, with freedom.

NICK SCHIFRIN: Linda Higuera is running to head the county administration in San Cristobal.

She's a member of the Green Party, aligned with President Lopez Obrador.

Earlier this month, assailants attacked her campaign car after she participated in an LGBTQ event.

LINDA HIGUERA (through translator): They were screaming homophobic words, hateful words, intimidating messages against me, saying: "You're not going to succeed.

You're a woman."

As a woman, you feel unprotected.

You're in an environment where you feel persecuted.

They destroy your van and there are no consequences.

NICK SCHIFRIN: In a violent state, ballots are guarded by soldiers.

But in some areas, armed groups blocked election workers from creating polling stations.

And in the indigenous town of Pantelo (ph), 500 people fled for fear of violence and the election won't even be held.

Claudia Rodriguez is the Chiapas executive director of the National Electoral Institute.

CLAUDIA RODRIGUEZ, National Electoral Institute (through translator): Obviously, it's because of the fear they have.

Some political actors have told us they don't want elections to take place.

It's difficult, and we as an institution support peaceful elections, but we cannot enter a community by force to elect a representative.

NICK SCHIFRIN: Chiapas' Green Party Chiapas' Green Party and Linda Higuera believe Claudia Sheinbaum can bring security.

LINDA HIGUERA (through translator): I'm sure she will be a great ally for all the women who need that support, that protection, so that we're no longer intimidated.

NICK SCHIFRIN: But opposition candidate Xochitl Galvez calls Lopez Obrador's security policy, known as hugs, not bullets, of failure.

XOCHITL GALVEZ, Mexican Opposition Presidential Candidate (through translator): Is security today better than ever?

Of course not; 186,000 people were murdered and 50,000 people disappeared.

That is the result of a security strategy, where the hugs have been for the criminals and the bullets for the citizens.

NICK SCHIFRIN: Sheinbaum's new solutions are a national intelligence agency and investments in young Mexicans who are vulnerable to organized crime.

LILA ABED: At the same time, she wants to continue consolidating the National Guard, wants to make sure it falls under the Ministry of Defense.

And so she would, in essence, continue what many have called the militarization of previous civilian forces in Mexico.

And there's a lot of concern, because the military is not necessarily trained on protecting human rights and making sure that when they are dealing with these new responsibilities, that they carry it out in a very responsible and legal manner.

NICK SCHIFRIN: The Mexican government has also used the military to block migrants' travel north, lowering migrant crossings, in cooperation with the Biden administration.

But it's not clear if that's sustainable, and the election has not produced alternative solutions.

LILA ABED: And Claudia Sheinbaum, at least she has said that she is going to ask the next U.S. president for funds to sustain the large amount of migrants that are now in Mexican territory.

But, more than that, there really is no comprehensive, detailed strategy or plan on behalf of one of the candidates or the other in terms of how they're going to deal with migration.

NICK SCHIFRIN: And so the woman whose family escaped Nazi persecution inherits major challenges throughout the country, but the glass ceiling won't be one of them.

For the "PBS NewsHour," I'm Nick Schifrin.