The aim of this textbook is to give an introduction to dierential geometry. It is based on the lectures given by the author at Eotvos Lorand University and at Budapest Semesters in Mathematics. In the rst chapter, some preliminary denitions and facts are collected, that will be used later. The classical roots of modern dierential geometry are presented in the next two chapters. Chapter 2 is devoted to the theory of curves, while Chapter 3 deals with hypersurfaces in the Euclidean space. In the last chapter, dierentiable manifolds are introduced and basic tools of analysis (dierentiation and integration) on manifolds are presented. At the end of Chapter 4, these analytical techniques are applied to study the geometry of Riemannian manifolds.