Based on lectures given by the authors at Moscow and Leningrad Universities, this textbook gives a systematic exposition of the main parts of Topology: Homology, Homotopy, Fibre Bundles, and Smooth Manifolds. The main purpose is to present all parts of topology as a unified whole. As a methodological innovation a coordinateless approach to the theory of fibre bundles and an exposition of the foundation of differerential topology without use of differential equations is presented. Many descriptive examples and drawings are included; each section is followed by a set of well chosen exercises. The reader needs only a basic knowledge of set theory, algebra and calculus; so first-year graduate students will find a thorough an profound introduction into the advanced theory of Differential Manifolds.