1. To stop talking about some topic. Would you please just shut up about politics while we're at the dinner table? I'm sick of hearing about it! Jack is so irritating. He never shuts up about how successful his business is.

2. To keep something secret; to not share or spread information about something. We all had to sign an agreement saying that we would shut up about the project we were working on. Tom and Mary aren't making their engagement public yet, so just shut up about it for the time being.