The first training course in homotopy topology and its numerous applications in Russian literature. Among the main topics covered in the book: the theory of cell complexes, homotopy groups, homology and cohomology, the method of spectral sequences, homotopy properties of manifolds. For the first time, the place and role of homotopy topology in modern mathematics and physics is described in a form accessible to a wide range of readers. The reader who has mastered the course will be able to freely navigate in the special scientific literature.