  We study a graph-theoretic model of interface dynamics called $Competitive\,
Erosion$. Each vertex of the graph is occupied by a particle, which can be
either red or blue. New red and blue particles are emitted alternately from
their respective bases and perform random walk. On encountering a particle of
the opposite color they remove it and occupy its position. We consider
competitive erosion on discretizations of `smooth', planar, simply connected
domains. The main result of this article shows that at stationarity, with high
probability, the blue and the red regions are separated by the level curves of
the Green function, with Neumann boundary conditions, which are orthogonal
circular arcs on the disc and hyperbolic geodesics on a general simply
connected domain. This establishes $conformal\,invariance$ of the model.
