  A Hamilton cycle in a graph $\Gamma$ is a cycle passing through every vertex
of $\Gamma$. A Hamiltonian decomposition of $\Gamma$ is a partition of its edge
set into disjoint Hamilton cycles. One of the oldest results in graph theory is
Walecki's theorem from the 19th century, showing that a complete graph $K_n$ on
an odd number of vertices $n$ has a Hamiltonian decomposition. This result was
recently greatly extended by K\"{u}hn and Osthus. They proved that every
$r$-regular $n$-vertex graph $\Gamma$ with even degree $r=cn$ for some fixed
$c>1/2$ has a Hamiltonian decomposition, provided $n=n(c)$ is sufficiently
large. In this paper we address the natural question of estimating $H(\Gamma)$,
the number of such decompositions of $\Gamma$. Our main result is that
$H(\Gamma)=r^{(1+o(1))nr/2}$. In particular, the number of Hamiltonian
decompositions of $K_n$ is $n^{(1-o(1))n^2/2}$.
