Although at rst glace an elliptic curve looks like a simple equation of the form y2 = x3 + ax + b, it has a rich structure and deep meaning, that open many interesting tools and questions in number theory. The essence of their uniqueness is the special group structure that one can easily give on the set of points of any elliptic curve. In this talk I will introduce this group structure and discuss its properties and related results such as the Mordell-Weil Theorem. Then I will brie y mention the cases of singular elliptic curves and elliptic curves over nite elds. Finally if time allows I will de ne and discuss the notion of complex multiplication and relate the properties of elliptic curve to some deeper results in number theory.