  The closure of a braid in a closed orientable surface $\Sigma$ is a link in
$\Sigma\times S^1$. We classify such closed surface braids up to isotopy and
homeomorphism (with a small indeterminacy for isotopy of closed sphere braids),
algebraically in terms of the surface braid group. We find that in positive
genus, braids close to isotopic links if and only if they are conjugate, and
close to homeomorphic links if and only if they are in the same orbit of the
outer action of the mapping class group on the surface braid group modulo its
center.
