Automorphisms of algebraic varieties and infinite transitivity

The paper reviews the results of recent years on the multiple transitivity of actions of the automorphism groups of affine algebraic varieties. The property of infinite transitivity for the action of the group of special automorphisms is considered and the equivalent flexibility property of the variety. These properties have important algebraic and geometric consequences, and at the same time they are fulfilled for wide classes of manifolds. The cases when infinite transitivity occurs for automorphism groups generated by a finite number of one-parameter subgroups are studied separately. In the appendices to the paper, the results on infinitely transitive actions in complex analysis and in combinatorial group theory are considered.