Moire superlattices enable the generation of new quantum phenomena in two-dimensional heterostructures, in which the interactions between the atomically thin layers qualitatively change the electronic band structure of the superlattice. For example, mini-Dirac points, tunable Mott insulator states and the Hofstadter butterfly pattern can emerge in different types of graphene/boron nitride moire superlattices, whereas correlated insulating states and superconductivity have been reported in twisted bilayer graphene moire superlattices(1-12). In addition to their pronounced effects on single-particle states, moire superlattices have recently been predicted to host excited states such as moire exciton bands(13-15). Here we report the observation of moire superlattice exciton states in tungsten diselenide/tungsten disulfide (WSe2/WS2) heterostructures in which the layers are closely aligned. These moire exciton states manifest as multiple emergent peaks around the original WSe2 A exciton resonance in the absorption spectra, and they exhibit gate dependences that are distinct from that of the A exciton in WSe2 monolayers and in WSe2/WS2 heterostructures with large twist angles. These phenomena can be described by a theoretical model in which the periodic moire potential is much stronger than the exciton kinetic energy and generates multiple flat exciton minibands. The moire exciton bands provide an attractive platform from which to explore and control excited states of matter, such as topological excitons and a correlated exciton Hubbard model, in transitionmetal dichalcogenides.