What does Don't wait up mean?

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To 'wait up' for someone, is to stay awake late into the night, waiting until you know they are back home again and made it home safely, before letting yourself go to sleep for the night.

Telling someone 'don't wait up' means "I know it will be very late when I get home, but I do not want you to worry. So just go to sleep at a normal time, and trust that I will get home safe. I should be fine, and it is not a sign of anything being wrong if I am not home until late tonight, because this is part of the expected plans for today that I will be home extra late at night tonight. So do not worry, and do not lose sleep worrying. I will be okay, even if I only get home at a very late time tonight."