If a US citizen marries a Mexican citizen in Mexico is the Mexican citizen eligible for a green card and travel to the US?

Whether your intended husband is able to obtain a green card and come to the US through your sponsorship after you are married depends upon his immigration and criminal history. He can obtain a copy of his immigration history through the Freedom of Information Act or a legal representative can do that for him. If he has a criminal history, he can obtain a copy of it through a legal representative. With the information, you and your intended husband can see whether there are any foreseeable obstacles to his immigration. If you have questions, you and he can engage an immigration lawyer to evaluate the materials.  Due to the limitations of the Lawyers.com Forums, Alan Lee & Arthur Lee, Esqs.’ (the "Firm") participation in responding to questions posted herein does not constitute legal advice, nor legal representation of the person or entity posting a question. No Attorney/Client relationship is or shall be construed to be created hereby. The information provided herein by the Firm is general, and requires that the poster obtain specific legal advice from an attorney. The poster shall not rely upon the information provided herein as legal advice nor as the basis for making any decisions of legal consequence.