  We study Fano varieties endowed with a faithful action of a symmetric group,
as well as analogous results for Calabi--Yau varieties, and log terminal
singularities. We show the existence of a constant $m(n)$, so that every
symmetric group $S_k$ acting on an $n$-dimensional Fano variety satisfies $k
\leq m(n)$. We prove that $m(n)> n+\sqrt{2n}$ for every $n$. On the other hand,
we show that $\lim_{n\to \infty} m(n)/(n+1)^2 \leq 1$. However, this asymptotic
upper bound is not expected to be sharp.
  We obtain sharp bounds for certain classes of varieties. For toric varieties,
we show that $m(n)=n+2$ for $n\geq 4$. For Fano quasismooth weighted complete
intersections, we prove the asymptotic equality $\lim_{n\to \infty}
m(n)/(n+1)=1$. Among the Fano weighted complete intersections, we study the
maximally symmetric ones and show that they are closely related to the
Fano--Fermat varieties, i.e., Fano complete intersections in $\mathbb P^N$ cut
out by Fermat hypersurfaces.
  Finally, we draw a connection between maximally symmetric Fano varieties and
boundedness of Fano varieties. For instance, we show that the class of
$S_8$-equivariant Fano $4$-folds forms a bounded family. In contrast, the
$S_7$-equivariant Fano $4$-folds are unbounded.
