If you have unlawfully stayed in the U.S. after the expiration of your visa, or otherwise entered and remained in the U.S. illegally, you might think that marriage to a U.S. citizen will automatically waive your unlawful stay (which, if it lasted more than 180 days, is a ground of inadmissibility) and allow you to go forward with a green card application. In fact, there is no such automatic waiver.
But there are two ways that an immigrant married to a U.S. citizen can potentially avoid the problem of inadmissibility based on unlawful presence, including by:
Adjustment of status is a process that a foreign citizen who has married a U.S. citizen can go through in order to become a permanent resident of the U.S. without leaving the country. And that's an excellent thing, because it's only after leaving the U.S. that someone becomes subject to inadmissibility based on unlawful presence.
Normally, however, adjustment of status is available only for foreign citizens who entered the U.S. legally--that is, using a visa, on the visa waiver program, or by similar means.
The adjustment procedure is not available for immigrants who entered the U.S. illegally, unless they are lucky enough to fall into a rare exception, based on laws that have since expired. For more information, see a lawyer or read Who Can Get a Green Card Through Adjustment of Status?
In cases where the foreign citizen has remained illegally in the U.S. for more than six months after April 1, 1997 and cannot adjust status or has already left the U.S., he or she cannot enter the U.S. on an immigrant visa, as the fiancé(e) or spouse of a U.S. citizen typically would. The person's prior illegal stay acts as a bar to obtaining such a visa.
However, a showing of extreme hardship to the U.S. citizen if the foreign national were denied admission to the U.S. can result in the waiver of the foreign citizen's previous unlawful presence in the U.S. and thus lead to the immigrant visa/green card being granted by the U.S. consulate. Extreme hardship typically involves some sort of medical, financial, educational, or other personal issue that is directly linked to the denial of the foreign citizen's admission to the United States.
There is a formal process to apply for this "unlawful presence waiver". The best option is to file what's called a "provisional waiver" to USCIS before leaving the U.S. for the consular interview. That allows you to receive an approval before taking the risk of departing for the consular interview. However, that procedure is open only to applicants who are not inadmissible on any grounds other than unlawful presence, and who are claiming extreme hardship to a U.S. citizen spouse or parent.
Applicants who do not meet the criteria for applying for a provisional waiver--or even whose provisional waiver request is denied--can take the chance of leaving the U.S. for their consular interview, being found inadmissible at that time, and then applying for a "regular" I-601 waiver.
You'll need to make a compelling argument and offer evidence to "prove" the hardship.
Immigration law is very specialized, and is constantly changing. Complex immigration issues, such as those dealing with the waiver of an unlawful stay in the United States, normally benefit greatly from the assistance of an experienced immigration attorney.