We have experimentally determined the energies of the ground and first four excited excitonic states of the fundamental optical transition in monolayer WS_{2}, a model system for the growing class of atomically thin two-dimensional semiconductor crystals. From the spectra, we establish a large exciton binding energy of 0.32 eV and a pronounced deviation from the usual hydrogenic Rydberg series of energy levels of the excitonic states. We explain both of these results using a microscopic theory in which the nonlocal nature of the effective dielectric screening modifies the functional form of the Coulomb interaction. These strong but unconventional electron-hole interactions are expected to be ubiquitous in atomically thin materials.