We define a notion of conformal equivalence for discrete surfaces (surfaces composed of euclidean triangles). For example, multiplying the lengths of all edges incident with a single vertex by the same factor is considered to be a conformal change of metric. It turns out that finding a conformally equivalent flat metric on a given discrete surface amounts to minimizing a globally convex functional on the space of all metrics. This functional involves the Lobachevski function (known in the context of computing the volume of hyperbolic tetrahedra). This is not an accident, since surprisingly the whole theory is stongly related to hyperbolic geometry. There are important practical applications of our method to Computer Graphics in the context of texture mapping.