This chapter presents an introduction to elliptic curve cryptography. Elliptic curves provide an important source of finite abelian groups in which cryptographic schemes relying on the hardness of the discrete logarithm problem (DLP) can be implemented. One important advantage of elliptic curve groups over other finite abelian groups such as the subgroups of the multiplicative groups of finite fields is the fact that in the elliptic case only generic algorithms—which have exponential complexity—are known for the DLP and this allows the use of smaller parameters, which is advantageous in restricted computing environments such as, for example, smart cards. Another advantage is that they provide an appropriate context for the development of identity-based cryptography.