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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 theaction of $\text{Aut}(X)$ is $m$-transitive for all $m$, the transitivitydegree is infinite. We compute the transitivity degree for all quasi-affinetoric varieties and for many homogeneous spaces of algebraic groups. Also wediscuss a conjecture and open questions related to this invariant.

Homogeneous algebraic varieties and transitivity degree