  With every nontrivial connected algebraic group $G$ we associate a positive
integer ${\rm gtd}(G)$ called the generic transitivity degree of $G$ and equal
to the maximal $n$ such that there is a nontrivial action of $G$ on an
irreducible algebraic variety $X$ for which the diagonal action of $G$ on $X^n$
admits an open orbit. We show that ${\rm gtd}(G)\leqslant 2$ (respectively,
${\rm gtd}(G)=\nobreak 1$) for all solvable (respectively, nilpotent) $G$, and
we calculate ${\rm gtd}(G)$ for all reductive $G$. We prove that if $G$ is
nonabelian reductive, then the above maximal $n$ is attained for $X=G/P$ where
$P$ is a proper maximal parabolic subgroup of $G$ (but not only for such
homogeneous spaces of $G$). For every reductive $G$ and its proper maximal
parabolic subgroup $P$, we find the maximal $r$ such that the diagonal action
of $G$ (respectively, a Levi subgroup $L$ of ~$P$) on $(G/P)^r$ admits an open
$G$-orbit (respectively, $L$-orbit). As an application, we obtain upper bounds
for the multiplicities of trivial components in some tensor product
decompositions. As another application, we classify all the pairs $(G, P)$ such
that the action of $G$ on $(G/P)^3$ admits an open orbit, answering a question
of {\sc M. Burger}.
