Written by a well-known number theorist with an extensive cryptographic background, this book interweaves the theory of elliptic curves with their applications, particularly to cryptography. A thorough treatment is given for the discrete logarithm problem for elliptic curves. Recent applications of the Weil pairing and the Tate-Lichtenbaum pairing are discussed. The text treats elliptic curves over finite fields, the rationals, and the complex numbers. The book makes elliptic curves accessible to a large audience, including senior-level undergraduate and graduate students in mathematics and computer science.