Yes, you can travel abroad as a green card holder — that's one of the many benefits of being a permanent resident. However, your trip must be temporary and you cannot remain outside the United States for more than 1 year.
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You can't leave the country until you have your approved travel document in hand, so you should expect to spend the 3–5 months after submitting your green card application in the United States. Theoretically, you could be out of the United States for as long as you have a travel document (one year plus renewals).
If you are a lawful permanent resident (green card holder), you may leave the United States multiple times and reenter, if you do not intend to stay outside the United States for 1 year or more.
The current Spain visa requirements for U.S. green card holders are: A completed visa application containing correct, honest information. A valid passport, which must be valid for at least three months after your proposed Schengen exit date.
The Renewing Immigration Provisions of the Immigration Act of 1929 would update the existing Registry statute so that an immigrant may qualify for lawful permanent resident status if they have lived in the U.S. continuously for at least seven years before filing an application for lawful permanent resident status and ...