This thesis explores the theory of elliptic curves with complex multiplication needed to prove Noam Elkies result that there are infinitely many supersingular primes for every elliptic curve defined over Q . The first chapter contains a brief overview of the basic theory of elliptic curves, including the topics of the j-invariant, isogenies, the endomorphism ring, the Tate module, elliptic curves over finite fields, good and bad reduction, and elliptic curves over C . In the second chapter we discuss elliptic curves defined over C with complex multiplication focusing on the action of the ring class group. In Chapter 3 we discuss the properties of the j-function and modular polynomials in order to prove that the j-invariants of elliptic curves with complex multiplication are algebraic integers and that these j-invariants generate ring class fields. In Chapter 4 we discuss ray class fields and give a brief summary of idelic class field theory in order to prove the main theorem of complex multiplication and some important results of Deuring. Finally in Chapter 5 we conclude with a proof that every elliptic curve defined over Q has infinitely many supersingular primes.