As with any nonimmigrant (temporary) visa, when your stay on F-1 (academic student) visa is over, you are expected to leave the United States. The law contains various provisions to prod you to do just that. Below, we'll discuss the legal consequences of failing to leave on time.
Unlike most visa entrants, students are not typically given a precise departure date. When they arrive in the U.S. on student visas, the border control officers normally make a "D/S" notation in the I-94 Arrival/Departure Record (accessible online at the CBP website).
The D/S notation stands for "duration of status." In other words, you're allowed to stay in the United States for as long as you are pursuing a full-time course of study or followup practical training and otherwise complying with the terms of your F-1 visa.
In addition, you may stay another 60 days in order to prepare for your departure from the U.S. or to transfer to another school. This is called a "grace period."
Pay no attention to the expiration date shown on the actual F-1 visa that you originally received at the U.S. consulate. That is simply the last date upon which you could use the visa to enter the United States, not the date by which you must leave the United States. This is a common source of confusion, but a visa is simply an entry document. The I-94 controls the length of your stay as a nonimmigrant. (For further discussion, see I-94 Date vs. U.S. Visa Expiration Date.)
Staying beyond the time allowed on a nonimmigrant visa is a violation of U.S. immigration laws and carries various consequences.
First off, if you overstay your permitted time by even by one day, your visa is automatically void. (See § 222(g) of the Immigration and Nationality Act or I.N.A. or 8 U.S.C. § 1202(g).) The practical consequence of this is that, if you had been intending to apply for an extension or change of your immigration status without leaving the United States, you will not be able to do so. You will instead have to go to a U.S. embassy or consulate outside the United States (most likely in your country of nationality) to apply for a new visa.
Even more serious consequences start to arise if you accrue six months or more of "unlawful presence" in the United States, as described next.
A nonimmigrant over the age of 18 who overstays an authorized stay on a U.S. visa ordinarily accrues what's known as "unlawful presence," an important concept within U.S. immigration law.
If you accrue unlawful presence of six months or more, then voluntarily leave the United States and apply to return (whether with a nonimmigrant visa or an immigrant visa/green card), you will be considered inadmissible and barred from reentry for three years. If you accrue unlawful presence of one year or more, you will face a bar on reentry of ten years. There are waivers available in certain limited circumstances, but they're not easy to get, and depend on having U.S. family members who would suffer extreme hardship if your visa were denied.
So, let's say that after hanging around in the United States for a year after your studies have ended, you receive a job offer from a U.S. employer that hopes to help you get a green card. You visit an attorney and discover that, because your visa is void, one application process is closed to you: You will not be allowed to adjust your status (apply for a green card) within the United States. What's more, if you try the alternative process, of applying for the green card at a U.S. consulate in your home country, you'll face the three-or ten-year time bar. The employer is unlikely to willingly wait another three or ten years for you to come back and start work!
An important question for people on student visas, however, is when their period of unlawful presence began. Given the absence of a date on your I-94, this is a determination that's not easily made. Therefore, the law says that unlawful presence on an F-1 visa begins when a Department of Homeland Security (DHS) official or an immigration judge actually declares you to be unlawfully present, regardless of the date you began violating your status. Consult an attorney for the latest on this; it's an issue that has come under scrutiny in recent years.
If you have stayed beyond the time permitted on an F-1 visa, but are hoping to find a way to remain in the United States legally, consult an experienced immigration attorney for a full analysis of your case and assistance with further applications.