Non-Hermitian eigenvalue problems arise in a number of important applications, including quantum chemistry and control theory. While several solution techniques are known for this class of problems, their efficiency in tackling large-scale eigenvalue computations on modern parallel computers may be limited. In this talk we present an algorithm that computes a subset of eigenpairs corresponding to the eigenvalues of a non-Hermitian matrix that are closest to a given shift. The algorithm represents a block iteration and relies on preconditioning to achieve fast convergence. The method does not perform shift-and-invert and has a modest memory requirement. We demonstrate the proposed approach on a number of practical problems.