The author of the article undertakes to thoroughly study the nature of the movement of antiglobalists, which movement he rates as one of the elements of civil society; moreover, perceives it as a forerunner of the arising global civil society. It would be more correct, I.B.Levin maintains, to call it the Movement for an alternative globalization. Demonstrating certain results yielded by the antiglobalists' initiatives, the author points to elaboration of a strategy of arranging the dialogue with the bearers of globalization processes, and search for mutually acceptable solutions, as indispensable for further development of the movement and for its efficiency. Russia has so far been avoiding the problems brought forth by the antiglobalists; however, the tendencies of the societal development are bound to result in the formation of global consciousness.