  In the first part of this paper we provide a self-contained introduction to
(regularized) perturbation determinants for operators in Banach spaces. In the
second part, we use these determinants to derive new bounds on the discrete
eigenvalues of compactly perturbed operators, broadly extending some recent
results by Demuth et al. In addition, we also establish new bounds on the
discrete eigenvalues of generators of $C_0$-semigroups.
