If you came to the United States on a visitor or tourist visa (B-2), you are expected to leave within a certain time. That time is shown on your Form I-94 Arrival/Departure Record, which you can access on the Customs and Border Protection (CBP) website.
However, if you believe you have some grounds upon which to obtain U.S. lawful permanent residence (a green card), you might hope to simply stay in the U.S. while awaiting your green card, even after you are supposed to leave. The bottom line is, you'd need to get your green card application in before your visitor stay expires.
We'll discuss here not only what normally happens when someone overstays a visa, but the rare situations in which people who have overstayed can successfully get a U.S. green card.
If you overstay the time permitted on a visa, that visa becomes legally void (automatically canceled) as of the date that you were scheduled to leave the U.S., unless you have already applied for an extension or otherwise begun the process of having your immigration status changed. (See I.N.A. § 222(g), 8 U.S.C. § 1201(g).)
By remaining in the U.S. without taking these steps first, you violate U.S. immigration laws, most likely accrue what's known as "unlawful presence," and can be removed or subjected to future penalties.
To reiterate, you cannot, in most cases, allow your permitted stay to expire and then file an extension or change of status application to get you a new nonimmigrant (temporary) status; or else an adjustment of status application, to get you U.S. lawful permanent residence (a green card).
What's more, you will need to have valid reasons for requesting the extended or new status. Filing a frivolous application gives you no right to remain in the United States.
The immigration penalties get particularly harsh if you overstay your permitted time on a U.S. visa and accrue unlawful presence of 180 days or more. In such a case, voluntarily leaving the United States and applying to return (whether on a visa or green card) would result in your being barred from reentry for three years. If your overstay is one year or more, you'll face of bar on reentry of ten years. There are waivers available in certain limited circumstances.
If you've identified a potential basis upon which to apply for legal immigration status in the United States, speak to an immigration lawyer immediately. Limited circumstances exist under which you could actually submit your application in the United States despite the overstay and record of unlawful presence and proceed to obtain a green card without having to leave the United States.
For example, someone who marries a U.S. citizen becomes an "immediate relative." This is one of the few categories of applicants allowed to apply for a green card within the United States through the process called adjustment of status (but it works ONLY for people who entered the U.S. lawfully, most likely using a visa; not for undocumented persons). Because applicants don't have to leave the U.S. during the adjustment of status process, they cannot then be penalized for a lengthy visa overstay with a three-or ten-year time bar.
Or, if your overstay hasn't yet reached six months, you could leave the United States before you hit that mark in order to preserve your right to apply to return. Again, talking to a lawyer about your strategy is your best bet.