You can do any of the following freely, without permission:
A work falls into the Public Domain because it is a publication of the U.S. federal government, or else its copyright has expired. The duration of copyright has changed several times. Currently it is:
Under some Creative Commons licenses, a work can be donated to the Public Domain, but it's unclear whether those licenses hold up under the Supreme Court. It's safest to just treat them as Creative Commons works licensed for Attribution.
Not actually in the Public Domain (unless other factors put it there):
Multimedia often contains multiple copyrights on the same work. A Shakespeare play is in the Public Domain because it is very old. But a movie based on it is not because the adaptation is a new derivative work with its own copyright. To make things more complicated, a song in this movie has two of its own copyrights - one for the composition and one for the performance.
If you want to use an old play or piece of music, you have to be careful that not only the original score/script is in the Public Domain, but also the recording of the performance. Otherwise you will need to take advantage of a different copyright exemption, or get permission from the copyright owner.
The rules for Public Domain are different (and generally more lenient) in the United States than in other countries. Works that are published only in the U.S. follow the U.S. rules, and so do works that are published in the U.S. and another country simultaneously. Works that are published only in another country, or that are published in another country before they are published in the U.S., follow the international rules.
If a work was published 1978 or later, use the Life + 70 or Creation + 120 rule. If the work was published internationally before 1890 or in the U.S. before
1923 1925 (as of 2020) then it is in the Public Domain.
Otherwise, the rules are more complicated. Here are some resources to help you figure it out.