Abstract: Recently, elliptic curve cryptosystems have become as a promising new area in public key cryptography. The main advantage of elliptic curve cryptosystems over other public key cryptosystems is the fact that they are based on a different intractable problem, and that their keys have smaller sizes for the same level of security. These properties lead to some better performance in application. Because of their security and efficiency, this kind of cryptosystems can be used widely in the future. In fact, they have come into strong consideration not only by theory researchers but also by standards developers. The paper here covers the main various aspects of elliptic curve cryptosystems. The concepts of elliptic curves and their related knowledge are introduced briefly, and a few typical schemes are given. The security of elliptic curve cryptosystems is given a properly detailed description. Building of an elliptic curve suitable for cryptosystems, which is essential to the setup of an elliptic curve cryptosystem, is discussed in detail. As a summary,the advance and the status quo of the elliptic curve cryptosystems are reflected, and so do the problems the cryptosystems face now. The newest achievements in this area are covered and views on the development of the cryptosystems are proposed.