A biperiodic planar network is a graph embedded on a torus and a function called conductance, that associates to each edge of the graph a non-zero complex number. The fundamental operator in the study of networks is the discrete Laplacian. Associated to the Laplacian of a biperiodic planar network is its spectral transform, a curve and a divisor on it. We provide a classification of biperiodic planar networks in terms of the spectral transform. The space of networks has a large group of automorphisms that arise from the Y-Delta move. We show that these automorphisms are linearized by the spectral transform. Speaker(s): Terrence George (Brown University)