Mulumbu - who can count West Bromwich Albion and Norwich City among his previous clubs - had been available on a free transfer after leaving Kilmarnock at the end of last season.
Mulumbu told the club's official website: "I walked on to the pitch when I signed and it felt amazing. These are the reasons we play football.
"My immediate focus is to stay fit and ready and be 100% ready. I've played at Celtic Park as the opposition and it's a special place to play, so I'm looking forward to being part of the team and playing there.
"I cannot describe how amazing this day is for me, but it's one step to the other. I have signed but now I must work hard to play."
Mulumbu is available to feature in the squad for the Old Firm derby against Rangers on Sunday.