The book is based on a topology course that was taught to first and second year NMU students, as well as American students as part of the Math in Moscow program. The first part is a general introduction to topology, with an emphasis on small geometric objects (graphs, surfaces, curves on a plane, nodes) and their invariants (Euler characteristic, degree of circle mapping, degree of point relative to the curve, fundamental group). The second part is an introduction to algebraic topology involving homotopy groups, cellular, simplicial and singular homologies, together with such classics as Poincaré duality, obstacle theory, Gurevich's theorems, Hopf-Whitney, Lefschetz, Eilenberg-MacLane spaces, vector bundles. For students and university professors.