  The central problem we consider is the distribution of eigenvalues of closed
linear operators which are not selfadjoint, with a focus on those operators
which are obtained as perturbations of selfadjoint linear operators. Two
methods are explained and elaborated. One approach uses complex analysis to
study a holomorphic function whose zeros can be identified with the eigenvalues
of the linear operator. The second method is an operator theoretic approach
involving the numerical range. General results obtained by the two methods are
derived and compared. Applications to non-selfadjoint Jacobi and Schr\"odinger
operators are considered. Some possible directions for future research are
discussed.
