The book provides an introduction to topology, differential topology, and differential geometry. After an introduction to fundamental concepts and results from set-theoretic topology, the Jordan curve theorem for polygonal chains is proven, giving a first idea of the nature of deeper topological problems. In the second chapter, manifolds and Lie groups are introduced and illustrated with a series of examples. Tangential and vector bundles, differentials, vector fields, and Lie brackets of vector fields are also discussed. This discussion is further deepened in the third chapter, where de Rham cohomology and oriented integrals are introduced, and the Brouwer fixed-point theorem, the Jordan-Brouwer separation theorem, and Stokes' integral theorem are proven. The concluding fourth chapter is dedicated to the foundations.