Let $t\mapsto A(t)$ for $t\in T$ be a $C^M$-mapping with values unbounded
operators with compact resolvents and common domain of definition which are
self-adjoint or normal. Here $C^M$ stands for $C^\om$ (real analytic), a
quasianalytic or non-quasianalytic Denjoy-Carleman class, $C^\infty$, or a
H\"older continuity class $C^{0,\al}$. The parameter domain $T$ is either
$\mathbb R$ or $\mathbb R^n$ or an infinite dimensional convenient vector
space. We prove and review results on $C^M$-dependence on $t$ of the
eigenvalues and eigenvectors of $A(t)$.