Yes. A US passport book is mandatory when traveling to Mexico by air. Also, minors (babies and children) will need a passport to enter Mexico. Once you arrive in the country, border control authorities will check and stamp your passport for entry.
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A visa, or visa stamp, is a physical stamp or sticker in your passport that is issued by a U.S. embassy or consulate outside of the U.S. It indicates that you are eligible to apply for entry to the U.S. in a specific immigration category such as F-1 or J-1 student.
You may have noticed that a visit to other countries implied a visa stamp on your passport, but this is not valid for Cuba. Your passport will not get stamped upon entry. Only the Tourist Card will receive a Cuba visa stamp, but you need to keep it in your passport at all times.
Warped, bent, or curved covers or pages. Peeling plastic. Unauthorized markings: some tourist attractions will have their own 'passport stamp', but since these are not official government agencies, they could. invalidate your passport.
I've read several posts that said the Post Office will stamp your passport. That's on the southern edge of downtown. If you port at Punta Langosta (the downtown pier) its right across the street.