  We survey recent results on multiple transitivity of automorphism groups of
affine algebraic varieties. We consider the property of infinite transitivity
of the special automorphism group, which is equivalent to flexibility of the
corresponding affine variety. These properties have important algebraic and
geometric consequences. At the same time they are fulfilled for wide classes of
varieties. Also we study situations where infinite transitivity takes place for
automorphism groups generated by finitely many one-parameter subgroups. In the
appendices to the paper, the results on infinitely transitive actions in
complex analysis and in combinatorial group theory are discussed.
