In Steklov eigenproblems for elliptic operators, the spectral parameter links boundary traces of eigenfunctions to traces of the Neumann data. It is natural, therefore, to reformulate such eigenproblems in terms of boundary integral operators, which allow for nonsmooth boundaries. In this talk we describe such strategies in the context of Steklov problems for the Laplacian as well as the Helmholtz operator, and their use in studying questions arising in spectral geometry.