When someone says “may not,” they're usually telling you that something isn't allowed or permitted. If your teacher says “You may not leave the classroom until the bell rings,” she means you don't have permission to go. If a sign at the pool says “Children may not swim without adult supervision,” it means swimming alone breaks the rules.

This phrase is different from “cannot” or “can't.” When you can't do something, it's impossible: you can't fly like a bird or breathe underwater. When you may not do something, you're physically able to do it, but the rules or someone in authority forbids it. You can run in the hallway, but you may not because it's against school rules.

“May not” can also express possibility, as in “It may not rain today,” meaning there's a chance it won't rain. But in that case, people often say “might not” instead.

Sometimes people use “may not” to sound more formal or polite than “can't.” A librarian might say “You may not eat in the library” rather than “You can't eat in the library,” even though both communicate the same rule.