Abstract: In this talk we shall discuss some recent progress on bounds of Keller and Lieb-Thirring type for eigenvalues of Schr\"odinger operators with complex potentials on non-trapping asymptotically conic manifolds. While in the self-adjoint case such bounds are classical, the non-self-adjoint case is considerably more involved and the sharpest results currently available were obtained quite recently by R. Frank and collaborators, in the Euclidean setting. We show that such results extend to the general setting of non-trapping asymptotically conic manifolds. A crucial ingredient in our proofs are weighted uniform estimates in suitable Schatten classes for the resolvent of the Laplacian on non-trapping asymptotically conic manifolds. This is joint work with Colin Guillarmou and Andrew Hassell.