Association des Annales de l'institut Fourier, 2015. In this paper, we develop a quantitative K-theory for filtered C-algebras. Particularly interesting examples of filtered C-algebras include group C-algebras, crossed product C-algebras and Roe algebras. We prove a quantitative version of the six term exact sequence and a quantitative Bott periodicity. We apply the quantitative K-theory to formulate a quantitative version of the Baum- Connes conjecture and prove that the quantitative Baum-Connes conjecture holds for a large class of groups.
