After an introduction to matrix measures, we discuss matrix polynomials in general as well as a broad class of examples. For one of these examples we discuss how one can build a set of exceptional matrix polynomials. This involves a set of orthogonal matrix polynomials, which are not directly labelled by their degree. These polynomials are constructed using a matrix Darboux factorisation. The last example is joint work with Pablo Roman and Lucia Morey.