  We develop a computational method for extremal Steklov eigenvalue problems
and apply it to study the problem of maximizing the $p$-th Steklov eigenvalue
as a function of the domain with a volume constraint. In contrast to the
optimal domains for several other extremal Dirichlet- and Neumann-Laplacian
eigenvalue problems, computational results suggest that the optimal domains for
this problem are very structured. We reach the conjecture that the domain
maximizing the $p$-th Steklov eigenvalue is unique (up to dilations and rigid
transformations), has p-fold symmetry, and an axis of symmetry. The $p$-th
Steklov eigenvalue has multiplicity 2 if $p$ is even and multiplicity 3 if
$p\geq3$ is odd.
