The Blues have become the much-travelled Anelka's eighth club but, at the age of 28, he feels ready to put down some roots.
"My time at Bolton was great and I want to thank everyone connected with the club for what they did for me. But this was too good an opportunity to turn down.
"I want to play Champions League football, I want to win trophies - and Chelsea can give me that.
"I'm really looking forward to working with Avram Grant and following his style of football because it suits my game. I've been really impressed by what he and the team have done this season in what were very difficult circumstances.
"They have had great results and are still challenging for all the trophies.
"The quality of players here is fantastic and the competition for places excites me."
Anelka has arrived at just the right time with top-scorer Didier Drogba in Ghana for the African Nations Cup.
The French international striker added: "I've played for big clubs before, I know what they are about and I know I can score goals at the top level of the game.