If someone from a European country has overstayed their J1 for over ten years and marries a citizen, what is the process to get a green card?

Generally, a U.S. citizen may sponsor a spouse to become a Lawful Permanent Resident (to get a "Green Card") in the immediate relative visa category if the foreign national spouse entered the U.S. lawfully and with inspection (such as with a J-1 visa), and this is true even if that spouse may have long overstayed the original visa. When a foreign national spouse has had a J-1 visa, however, it will be necessary to address whether the person was subject to a two-year home residency requirement. If such a requirement was associated with the J-1 visa, then it will need to be satisfied or the requirement will need to be formally waived before the person becomes eligible for adjustment of status. It would be wise for the couple to consult with an immigration attorney who, after learning all of the relevant information, including details about the J-1 visa, could advise about immigration eligibilities, options and strategies, and could offer legal representation in the often complex application process.