Moire superlattices provide a powerful tool to engineer novel quantum phenomena in two-dimensional (2D) heterostructures, where 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 can emerge in different types of graphene/boron nitride moire superlattices, while correlated insulating states and superconductivity have been reported in twisted bilayer graphene moire superlattices. In addition to their dramatic effects on the single particle states, moire superlattices were recently predicted to host novel excited states, such as moire exciton bands. Here we report the first observation of moire superlattice exciton states in nearly aligned WSe2/WS2 heterostructures. 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 distinctly different from that of the A exciton in WSe2 monolayers and in large-twist-angle WSe2/WS2 heterostructures. The observed phenomena can be described by a theoretical model where the periodic moire potential is much stronger than the exciton kinetic energy and creates multiple flat exciton minibands. The moire exciton bands provide an attractive platform to explore and control novel excited state of matter, such as topological excitons and a correlated exciton Hubbard model, in transition metal dichalcogenides.