The study of the arithmetic of elliptic curves is an extremely rich area of mathematics, incorporating ideas from complex analysis, algebraic geometry, group and Galois theory, and of course number theory. Many fundamental problems in number theory, such as Fermat's Last Theorem and the congruent number problem, lead naturally to the study of elliptic curves. The purpose of this course is to explore some of the basic ideas in this area, including the group law on elliptic curves, the structure of this group over various fields, and the Birch and Swinnerton-Dyer conjeture.