  We obtain asymptotic formulae for the Steklov eigenvalues and eigenfunctions
of curvilinear polygons in terms of their side lengths and angles. These
formulae are quite precise: the errors tend to zero as the spectral parameter
tends to infinity. The Steklov problem on planar domains with corners is
closely linked to the classical sloshing and sloping beach problems in
hydrodynamics; as we show it is also related to quantum graphs. Somewhat
surprisingly, the arithmetic properties of the angles of a curvilinear polygon
have a significant effect on the boundary behaviour of the Steklov
eigenfunctions. Our proofs are based on an explicit construction of quasimodes.
We use a variety of methods, including ideas from spectral geometry, layer
potential analysis, and some new techniques tailored to our problem.
