Estimates for eigenvalues of Schr\"{o}dinger operators with complex-valued potentials

New estimates for eigenvalues of non-self-adjoint multi-dimensional Schr"{o}dinger operators are obtained in terms of $L_p$-norms of the potentials. The results extend and improve those obtained previously. In particular, diverse versions of an assertion conjectured by Laptev and Safronov are discussed. Schr"{o}dinger operators with slowly decaying potentials are also considered.