This book surveys some recent developments in the arithmetic of modular elliptic curves. It places special emphasis on the construction of rational points on elliptic curves, the Birch and Swinnerton-Dyer conjecture, and the crucial role played by modularity in shedding light on these two closely related issues. The main theme of the book is the theory of complex multiplications, Heegner points, and some conjectural variants. Author information is not given. Material is based on a lecture series given by the author at the CBMS Conference on Modular Elliptic Curves, held at the University of Central Florida, August 2001.