Moiré 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 moiré superlattices, whereas correlated insulating states and superconductivity have been reported in twisted bilayer graphene moiré superlattices. In addition to their pronounced effects on single-particle states, moiré superlattices have recently been predicted to host excited states such as moiré exciton bands. Here we report the observation of moiré superlattice exciton states in tungsten diselenide/tungsten disulfide (WSe/WS) heterostructures in which the layers are closely aligned. These moiré exciton states manifest as multiple emergent peaks around the original WSe A exciton resonance in the absorption spectra, and they exhibit gate dependences that are distinct from that of the A exciton in WSe monolayers and in WSe/WS heterostructures with large twist angles. These phenomena can be described by a theoretical model in which the periodic moiré potential is much stronger than the exciton kinetic energy and generates multiple flat exciton minibands. The moiré 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 transition-metal dichalcogenides.