We will present a new method to obtain upper bounds on the number of eigenvalues of linear operators on Banach spaces. More precisely, we will consider linear operators $L=L_0+K$ which arise from some free operator $L_0$ by a compact perturbation $K$ and derive bounds on the number of eigenvalues of $L$ in the complement of the spectrum of $L_0$. This talk is based on joint work with M. Demuth, F. Hanauska and G. Katriel.