Over the past 20 years, elliptic curves have become an integral part of many cryptographic protocols. As a consequence, methods for constructing elliptic curves suitable for cryptography have been studied intensively. This book gives a new and very efficient method to construct an elliptic curve of prescribed order. It includes a basic introduction to elliptic curves, written for beginning graduate students in mathematics. The presented construction naturally leads into a discussion of the relationship between elliptic curves and algebraic number theory. This more advanced topic, known as complex multiplication theory, was first pioneered by Kronecker in the 19th century and it has recently regained popularity. The second part of this book contains various new p-adic algorithms in this area and it gives a description of p-adic class invariants.