  Let $X$ be an algebraic variety such that the group $\text{Aut}(X)$ acts on
$X$ transitively. We define the transitivity degree of $X$ as a maximal number
$m$ such that the action of $\text{Aut}(X)$ on $X$ is $m$-transitive. If the
action of $\text{Aut}(X)$ is $m$-transitive for all $m$, the transitivity
degree is infinite. We compute the transitivity degree for all quasi-affine
toric varieties and for many homogeneous spaces of algebraic groups. Also we
discuss a conjecture and open questions related to this invariant.
