A linear code over a finite field or ring R is called a maximum weight spectrum (MWS) code if it has the maximum number $s$ of distinct (nonzero) weights among all linear codes of a fixed module type. The underlying weight function is usually taken as the Hamming weight if R is a finite field and as the homogeneous weight if R is a finite chain ring. In these cases $s$ has been determined completely. In my talk I will explain these facts and also discuss recent surprising results about the smallest possible lengths of MWS codes.