The aim of our stay at the CIRM is to investigate the relationship between the geometry of a Euclidean domain and the spectrum of the associated Laplacian, subject to complex Robin boundary conditions. The Robin Laplacian has attracted a lot of attention in recent years, but the majority of the results restrict to real-valued boundary functions. Motivated by non-conservative physical systems, where the imaginary part of the boundary function models dissipative effects, and quasi-Hermitian quantum mechanics, our goal is to fill this gap by developing a spectral theory of the Robin Laplacian in the general nonself-adjoint setting. We are particularly interested in the asymptotic behaviour of the eigenvalues and eigenfunctions in the limit of strong boundary coupling, where the geometry of the domain is known to play an important role in the self-adjoint case. The strategy we suggest involves a development of semiclassical techniques for Schrödinger operators on manifolds with complex-valued potentials, a careful analysis of the resolvent of the Robin Laplacian and Agmon-type estimates for the eigenfunctions.