  We introduce new geometric objects called spectral networks. Spectral
networks are networks of trajectories on Riemann surfaces obeying certain local
rules. Spectral networks arise naturally in four-dimensional N=2 theories
coupled to surface defects, particularly the theories of class S. In these
theories spectral networks provide a useful tool for the computation of BPS
degeneracies: the network directly determines the degeneracies of solitons
living on the surface defect, which in turn determine the degeneracies for
particles living in the 4d bulk. Spectral networks also lead to a new map
between flat GL(K,C) connections on a two-dimensional surface C and flat
abelian connections on an appropriate branched cover Sigma of C. This
construction produces natural coordinate systems on moduli spaces of flat
GL(K,C) connections on C, which we conjecture are cluster coordinate systems.
