This project aims to describe Pollard's rho-Algorithm for solving the Discrete Logarithm Problem in a group, and to look at how this algorithm applies to Elliptic Curve Cryptography. We start with a general definition of Elliptic Curves and the Discrete Logarithm Problem, and go on to describe Pollard's rho-algorithm in detail. We prove some of the assumptions used in the algorithm description, and in the final section we look at some of the issues with which one is faced when trying to apply the algorithm to the Elliptic Curve Discrete Logarithm Problem.