I overstayed in the US, and now I want to go back to my home country (India) after three years.
Will there be a problem with US Immigration while departing?
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You will not have any issues when leaving the US. The US doesn't have physical immigration when you are leaving the country - it's all done electronically.
Once you check-in (and/or no later than one hour before the flight) the airline will pass your details to the CBP, who will detect the overstay, but given that at that point you're on the way out of the country they will not attempt to do anything about it at that time.
However once you leave, you can pretty much plan on never coming back, or at least not for a very long time. An overstay of more than 1 year will earn you an automatic 10 year ban from entering the US. Your existing visa (if it's still valid) will be invalidated, and you will not be able to obtain a new one for (at least) 10 years - with very ltitle recourse to appeal that fact. Even once your 10 years is up it's going to be difficult to convince the consulate/border staff that you will not overstay again if you are let into the country.
(There are a few exceptions to this, but likely the only one that's relevant is if you are under the age of 18, but I'm guessing this isn't the case)
Note that if you don't leave voluntarily and you are caught and deported, then the punishment is even more severe - so leaving on your own is still a good thing!
The US has no exit immigration, but you will definitely have a hard time getting a new visa, or, if it's still valid, it will be revoked when your exit is electronically recorded. If you then try travelling to the US again, the check-in staff will receive a "DO NOT BOARD" message after you submit your APIS info.
If you need to visit the US again, apply for a waiver of inadmissibility at the US embassy well in advance of your trip.
If you plan on returning to the USA at some point in your life (and this includes even transfers, flights to, say, South America where you are merely connecting in the USA), you may want to consult with an immigration attorney before you leave.
The 10-year ban others have mentioned is real, but it does not actually kick in until you physically leave. It also only applies if you have been unlawfully present for a year or more (the ban is 3 years if you have six months or more, and there is no automatic ban if the overstay was shorter). There are, of course, additional consequences; your visa will be invalidated, and you will find it difficult to get a new one even if no formal ban applies.
Before you leave, a good immigration attorney may have some strategies available to solve that issue. I can think of two strategies that may or may not work, depending on your situation.
One is through marriage to a US citizen, then applying for a Green Card. Once you have a Green Card, you can freely travel in and out of the USA.
Another is supposedly to not leave on your own, but rather trigger deportation proceedings. The 10-year ban does not apply when you were deported. Of course, deportation carries its own consequences (which can include a 5-year ban, a 20-year ban or a lifetime ban). Now the trick is that you can avoid the consequences of a deportation by using what is called voluntary departure. AFTER deportation proceedings have started, you can apply for voluntary departure - meaning that you save the US government the expense of deportation.