  Let $G$ be a graph without isolated vertices. The total domination number of
$G$ is the minimum number of vertices that can dominate all vertices in $G$,
and the paired domination number of $G$ is the minimum number of vertices in a
dominating set whose induced subgraph contains a perfect matching. This paper
determines the total domination number and the paired domination number of the
toroidal meshes, i.e., the Cartesian product of two cycles $C_n$ and $C_m$ for
any $n\ge 3$ and $m\in\{3,4\}$, and gives some upper bounds for $n, m\ge 5$.
