Book excerpt: Invited to translate his 1992 book from the Russian, Chigogidze chose instead to produce an updated version. He introduces a powerful method used in various branches of topology and also being applied to functional analysis and algebra. He surveys the Hilbert cube and Hilbert space manifold theories, recent developments of the Menger and Nobeling manifold theories, infinite-dimensional manifolds, cohomological dimensions, the general theory of absolute extensors in dimension n and n-soft mappings, the topology of non-metrizable manifolds, and applications in a number of areas. Annotation copyrighted by Book News, Inc., Portland, OR