  A \emph{domatic} (\emph{total domatic}) \emph{$k$-coloring} of a graph $G$ is
an assignment of $k$ colors to the vertices of $G$ such that each vertex
contains vertices of all $k$ colors in its closed neighborhood (neighborhood).
The \emph{domatic} (\emph{total domatic}) \emph{number} of $G$, denoted $d(G)$
($d_t (G)$), is the maximum $k$ for which $G$ has a domatic (total domatic)
$k$-coloring. In this paper, we show that for two non-trivial graphs $G$ and
$H$, the domatic and total domatic numbers of their Cartesian product $G \cart
H$ is bounded above by $\max\{|V(G)|, |V(H)|\}$ and below by $\max\{d(G),
d(H)\}$. Both these bounds are tight for an infinite family of graphs. Further,
we show that if $H$ is bipartite, then $d_t(G \cart H)$ is bounded below by
$2\min\{d_t(G),d_t(H)\}$ and $d(G \cart H)$ is bounded below by
$2\min\{d(G),d_t(H)\}$. These bounds give easy proofs for many of the known
bounds on the domatic and total domatic numbers of hypercubes \cite{chen,zel4}
and the domination and total domination numbers of hypercubes \cite{har,joh}
and also give new bounds for Hamming graphs. We also obtain the domatic (total
domatic) number and domination (total domination) number of $n$-dimensional
torus $\mathop{\cart}\limits_{i=1}^{n} C_{k_i}$ with some suitable conditions
to each $k_i$, which turns out to be a generalization of a result due to
Gravier \cite{grav2} %[\emph{Total domination number of grid graphs}, Discrete
Appl. Math. 121 (2002) 119-128] and give easy proof of a result due to
Klav\v{z}ar and Seifter \cite{sand}.
