  The paper is concerned with the Steklov eigenvalue problem on cuboids of
arbitrary dimension. We prove a two-term asymptotic formula for the counting
function of Steklov eigenvalues on cuboids in dimension d greater or equal to
3. Apart from the standard Weyl term, we calculate explicitly the second term
in the asymptotics, capturing the contribution of the (d-2)-dimensional facets
of a cuboid. Our approach is based on lattice counting techniques. While this
strategy is similar to the one used for the Dirichlet Laplacian, the Steklov
case carries additional complications. In particular, it is not clear how to
establish directly the completeness of the system of Steklov eigenfunctions
admitting separation of variables. We prove this result using a family of
auxiliary Robin boundary value problems. Moreover, the correspondence between
the Steklov eigenvalues and lattice points is not exact, and hence more
delicate analysis is required to obtain spectral asymptotics. Some other
related results are presented, such as an isoperimetric inequality for the
first Steklov eigenvalue, a concentration property of high frequency Steklov
eigenfunctions and applications to spectral determination of cuboids.
