Arithmetic properties of automorphic $L$-functions

In the first part of the talk, I will discuss the example of critical values for $L$-functions for $GL(2)$ in some detail. This part is an exposition, in the framework of automorphic representations, of the classical results of Manin, Shimura, etc. In the latter part, I will sketch results of my work, which is partly in separate collaborations with Grobner, Harder and Shahidi, on the special values for automorphic $L$-functions for higher $GL(n)$.