The aim of this course is to introduce the Elliptic Curves in Cryptography. After introducing the basic facts about the elliptic curves, we shall discuss the implementation of Elliptic Curves and Algorithms to compute group order. The emphasis will be given the elliptic curve cryptosystems and the related algorithms . The other applications of Elliptic Curves in Cryptography  such as primality and factorization tests will also be discussed.