Two lines of attack in the spectral theory of n x n matrix polynomials of degree d will be outlined. The first is an algebraic approach based on the notion of isospectral linear systems in $C^dn$ (the linearizations) and the second on analysis of associated matrix-valued functions acting on $C^n$. The first approach leads to canonical forms for real symmetric systems consisting of real matrix triples, and thence to canonical triples. Furthermore, for real selfadjoint systems we describe selfadjoint canonical triples of real matrices and illustrate their properties. It turns out that, in this context, there is a fundamental orthogonality property associated with the spectrum. It will be shown how this can play a role in inverse (spectral) problems, i.e. constructing systems with prescribed spectral properties.