If you have overstayed the time permitted under an F-1 academic student visa (or any other visa) before leaving the United States, you could have trouble returning to the U.S., even for a short visit. As discussed below, you face two potential problems:
First, let's check on the length of your actual permitted U.S. stay. Even if the expiration date on your F-1 student visa hasn't run out, that visa is just an entry document. You must look to your I-94, a document created by Customs and Border Protection (CBP) when you arrived at the U.S. airport, border, or other entry point, to see when your permitted stay ends. If you remain in the U.S. beyond that allowed time, your visa is automatically canceled.
There's a catch, though. You are likely to see "D/S" rather than a date on your I-94. That's because the student visa is an unusual one, in that when students enter the U.S., they are normally not given a firm departure date. Instead, their I-94 will say "D/S," for duration of status. That means you are allowed to stay in the U.S. until you have completed your studies (assuming you don't violate the terms of your status up until that point, for example by dropping out of school for a semester).
As a result, your overstay does not technically count as "unlawful presence" for purposes of the inadmissibility bars unless and until U.S. Citizenship and Immigration Services (USCIS) or an immigration judge has officially determined that your authorized stay is terminated as of a certain date.
Staying a few days or weeks past the date you were expected to leave the U.S. won't make you automatically inadmissible (ineligible for another visa or green card or U.S. entry). However, if your overstay lasted 180 days or more, you will have more serious inadmissibility issues, as follows:
Unlawful presence is a complex concept within U.S. immigration law, and various exceptions apply. For example, time spent in the U.S. unlawfully when one was under the age of 18 doesn't count. Some green card applicants might also be able to apply for a waiver (legal forgiveness) of their overstay.
For more on this topic, see Three-Year and Ten-Year Time Bars for Unlawful U.S. Presence.
After an overstay in the United States, you will need to apply for a new visa before returning to the country—and you will be restricted to making the application in your country of nationality, never in a third country (for example, if you're traveling and want to add the U.S. to your itinerary).
If you apply for another student visa, a visitor visa, or any other nonimmigrant (temporary) visa to the United States, you will most likely be asked to prove "nonimmigrant intent"—that is, that you truly plan to stay only for the time allotted, and not to attempt to settle in the United States. Your past visa overstay could, depending on its length and factual circumstances, raise doubts about your intent.
If you or a member of your family has overstayed the time permitted on an F-1 student visa and is having trouble obtaining another visa to the United States, it might be helpful to discuss the problem with an immigration attorney. The lawyer can review the circumstances of the visa overstay and assist you in understanding any options to rectify the situation.