A rapid development of K-theory of group $C^*$-algebras started about 40 years ago. To a large extent this was related with applications to topology of smooth manifolds (the Novikov conjecture on higher signatures) and to representation theory of semisimple Lie groups (and discrete discrete series in particular). But the topic of K-theory for group $C^*$-algebras is very attractive also by its deep relations with index theory of elliptic operators. There are several unsolved, very challenging conjectures in this theory, such as the Baum-Connes conjecture and the Kadison-Kaplansky conjecture. I will try to give a survey of the current status of this theory, without going into deep technical details. This talk will be accessible to graduate students. (Some knowledge of Hilbert space theory will be required).