The Laplacian on planar graphs and graphs on surfaces

These are lecture notes for the Current Developments in Mathematics conference at Harvard, November, 2011. We discuss topological, probabilistic and combinatorial aspects of the Laplacian on a graph embedded on a surface. The three main goals are to discuss: (1) for "circular" planar networks, the characterization due to Colin de Verdi\`ere of Dirichlet-to-Neumann operator; (2) The connections with the random spanning tree model; and (3) the characteristic polynomial of the Laplacian on an annulus and torus.