If your passport containing a valid U.S. visa has been lost or stolen, the first thing you should do is report the matter to the government agency in your home country that's responsible for issuing passports. If the passport was stolen, you should also file a police report. Many countries share lost and stolen passport data through Interpol and most passports now have a machine-readable chip that is scanned by immigration authorities, so if someone tries to use the passport, in any country, they will know that the person is an impostor.
Unfortunately, the Department of State does not issue a "replacement" for a lost or stolen U.S. visa. You will need to apply for the visa again. This policy is the same for every visa type, including tourist visas, student visas, and work visas.
Additionally, you must appear for an in-person interview, in the same manner that you did for your original visa, because you will not be eligible for an interview waiver without the physical visa and passport. While this will be a new application, these interviews are typically more like an interview for a visa renewal, except that the officer will want to see evidence that you reported the loss or theft of the passport and might ask you questions about the circumstances of the loss or theft.
If you lost your visitor visa, during the interview the officer will review your U.S. travel history and might ask questions about your trips. You should also bring proof of employment and your bank statements in case the officer wants to see them. Unlike first-time applications though, many people whose visitor visas are lost or stolen will apply for a replacement visa without any definite U.S. travel plans, just because they want to replace the visa. It's okay if you don't have any plans to travel to the U.S. and just want to get the new visa to avoid delays next time you travel. Still, the officer will want to make sure there have been no major changes in your life circumstances that might affect your eligibility for the visa.
If you lost a student visa or work visa, you will need to present the same documents that you did at your first interview. If you lost your passport and visa while you were back in your home country on a visit, you might be eligible for an expedited appointment. Each consulate has a procedure for these types of appointments, which you'll find on the consulate's website.
When you receive your new visa, take extreme precautions not to lose it again. If you have had more than one U.S. visa lost or stolen, you will likely encounter further scrutiny in your interview. The officer might limit the duration of your new visa, or might refuse your visa if there are any suspicions that you were complicit in the "loss" or theft. This is because there have been incidents where individuals sold, rented, or permitted family members or friends to borrow their passports with U.S. visas. This can be grounds for permanent visa ineligibility and you could even face criminal prosecution.
Once your visa and passport are reported lost or stolen, that information might be shared with immigration authorities around the globe and entered into databases that are accessible by airlines and governmental agencies Therefore, when traveling on your new visa, you might receive extra screening from airline personnel and immigration authorities, in order to verify your identity—in other words, to make sure you aren't actually the thief, posing as you.
When you travel, carry a copy of any reports you filed regarding your lost and stolen passport, including police reports. In the U.S., your identity can be verified with your fingerprint by U.S. immigration authorities, but you still might be asked about the circumstances that led to your old visa being lost or stolen.
If you recover your passport with a U.S. visa in it, and the visa was previously reported lost or stolen, you should not use it to travel to the United States.
Furthermore, some countries share lost and stolen passport data with the U.S. and this might be cross-referenced with U.S. visa issuances, resulting in the automatic cancellation of your visa, even if you did not report the visa as lost or stolen to the Department of State. This means that even if the authorities in your country have a way to reinstate a passport that was previously reported as lost, your U.S. visa inside the passport might no longer be valid or might be flagged for extra screening because it was associated with a lost or stolen passport.
The U.S. consulate in your home country might be able to verify the validity of your visa, but this does not guarantee that you will not be subject to additional screening at the U.S. port of entry.