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One of the best ways to learn topology is to read the classics. This translation of the 2005 Russian edition include the original methods and constructions from the originals of the 1950s and 1960s. The extended entries include Pontrjagin's article on smooth manifolds and their application in homotopy theory; Thoms' work on global properties of differential manifolds; Novikov's paper on homotropy properties of Tom complexes; papers by Smale on the generalized Poincare's conjecture in dimensions greater than four and the structure of manifolds; Quillen's article on the formal group laws of unoriented and complex cobordism theory; Buchstaber, Mischenko and Novikov's joint effort on formal groups and their role in algebraic topological approaches; and Buchstaber and Novikov's work on formal groups, power systems and Adams operators. Milnor's lectures on h-cobordism, which were in the original edition, have been omitted.