The spectra of elliptic operators are intricately connected to the geometrical properties of the spatial domains on which the operators are defined. Numerical computations are invaluable in studying this interplay, and high-accuracy discretizations are needed. This is particularly true of the Steklov problems. In this talk we'll present strategies for computing Steklov-Laplace and Steklov-Helmholtz spectra based on integral operators, and their efficacy in solving questions on the impact of geometry on spectral asymptotics. If time permits, we'll also present work in progress on a (modification of) the Steklov-Maxwell problem.