  The domatic number of a graph $G$, denoted $dom(G)$, is the maximum possible
cardinality of a family of disjoint sets of vertices of $G$, each set being a
dominating set of $G$. It is well known that every graph without isolated
vertices has $dom(G) \geq 2$. For every $k$, it is known that there are graphs
with minimum degree at least $k$ and with $dom(G)=2$. In this paper we prove
that this is not the case if $G$ is $k$-regular or {\em almost} $k$-regular (by
``almost'' we mean that the minimum degree is $k$ and the maximum degree is at
most $Ck$ for some fixed real number $C \geq 1$). In this case we prove that
$dom(G) \geq (1+o_k(1))k/(2\ln k)$. We also prove that the order of magnitude
$k/\ln k$ cannot be improved. One cannot replace the constant 2 with a constant
smaller than 1. The proof uses the so called {\em semi-random method} which
means that combinatorial objects are generated via repeated applications of the
probabilistic method; in our case iterative applications of the Lov\'asz Local
Lemma.
