The book covers standard material of the theory of elliptic curves. In the first six chapters, the Mordell theorem on the finite generation of rational points on elliptic curves defined over rational numbers is proved by elementary methods. This part grew out of Tate's 1961 Haverford Philips Lectures. The next part, consisting of Chapters 7 and 8, surveys Galois theory and then recasts arguments used in the proof of the Mordell theorem into the context of Galois representations and descent theory. The remaining series of sections contains an introduction to scheme theoretic properties of classifying spaces for families of elliptic curves. The topics include l-adic representations, L-functions, Birch and Schwinnerton-Dyer conjecture, etc. The book ends with three appendices containing applications of Calabi-Yau manifolds in string theory, applications of elliptic curves in cryptography and in a study of spectra of topological modular forms. The book contains a number of misprints (e.g., on page 180, the integral formula for the Γ- function should contain exp(-t2) instead of exp(-1), and on page 190, there should be π’s in the exponentials instead of n’s).