Generalizing the intriguing phenomenon of Voiculescu's almost commuting unitary matrices is the notion of a quasi-representation of a (discrete) group. As demonstrated by the work of Exel and Loring on Voiculescu's example, there may be topological obstructions to perturbing quasi-representations into genuine representations---this is where (topological or operator) K-theory enters the picture. Previous studies have mostly focused on fundamental groups of finite CW complexes. In this talk, which is based on joint work with Shmuel Weinberger and Guoliang Yu, we introduce the notion of a character as a more general and refined invariant for quasi-representations.