In this chapter we discuss some classes of endomorphisms (or square matrices) whose eigenvalues and eigenvectors have special properties. Such properties only exist under further assumptions, and in this chapter our assumptions concern the relationship between the given endomorphism and its adjoint endomorphism. Thus, we focus on Euclidean or unitary vector spaces. This leads to the classes of normal, orthogonal, unitary and selfadjoint endomorphisms. Each of these classes has a natural counterpart in the set of square (real or complex) matrices.