  An autotopism of a Latin square is a triple $(\alpha,\beta,\gamma)$ of
permutations such that the Latin square is mapped to itself by permuting its
rows by $\alpha$, columns by $\beta$, and symbols by $\gamma$. Let
$\mathrm{Atp}(n)$ be the set of all autotopisms of Latin squares of order $n$.
Whether a triple $(\alpha,\beta,\gamma)$ of permutations belongs to
$\mathrm{Atp}(n)$ depends only on the cycle structures of $\alpha$, $\beta$ and
$\gamma$. We establish a number of necessary conditions for
$(\alpha,\beta,\gamma)$ to be in $\mathrm{Atp}(n)$, and use them to determine
$\mathrm{Atp}(n)$ for $n\le17$. For general $n$ we determine if
$(\alpha,\alpha,\alpha)\in\mathrm{Atp}(n)$ (that is, if $\alpha$ is an
automorphism of some quasigroup of order $n$), provided that either $\alpha$
has at most three cycles other than fixed points or that the non-fixed points
of $\alpha$ are in cycles of the same length.
