We study eigenvalue asymptotics for a class of Steklov problems, possibly mixed with Dirichlet and/or Neumann boundary conditions, on planar domains with piecewise smooth boundary and with finitely many corners. This includes the famous "sloshing problem" as well as the Steklov problem on polygons. Two interesting features of this problem, which I will explain, are the surprisingly precise asymptotics we can obtain (with error decreasing as the spectral parameter increases) and a connection to a scattering problem on the Steklov portion of the boundary. This is joint work with M. Levitin (Reading), L. Parnovski (UCL), and I. Polterovich (Montreal).