Abstract: We will introduce Terry Lyons theory of rough paths. This theory allows pathwise integration against paths of arbitrary holder continuity. In particular this allows a pathwise theory of stochastic integration. When the driving path has Holder continuity less than or equal to 1/2, there are non-unique definitions for the integral that give a complete theory. We will explore how numerical analytic methods such as the Runge-Kutta integration scheme can define new integrals in this setting.