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1. To wrap something, such as a gift. Don't forget to wrap up Dan's present before you go to the party.
2. To complete something, such as a task; to put the finishing touches on something. You go on ahead, I just need to wrap up some work before I head home. OK, everyone, let's wrap it up for tonight.
*wrapped up (with someone or something)
involved with someone or something. (*Typically: be ~; get ~.) She is all wrapped up with her husband and his problems. She is just too wrapped up.
wrapped up (in someone or something)
concerned or obsessed with someone or something. Sally is pretty wrapped up in herself. I'm too wrapped up in my charity work to get a job.