  We prove two explicit bounds for the multiplicities of Steklov eigenvalues
$\sigma_k$ on compact surfaces with boundary. One of the bounds depends only on
the genus of a surface and the index $k$ of an eigenvalue, while the other
depends as well on the number of boundary components. We also show that on any
given smooth Riemannian surface with boundary, the multiplicities of Steklov
eigenvalues $\sigma_k$ are uniformly bounded in $k$.
