Quantum particles on a lattice with competing long-range interactions are ubiquitous in physics; transition metal oxides1,2, layered molecular crystals3 and trapped-ion arrays4 are a few examples. In the strongly interacting regime, these systems often show a rich variety of quantum many-body ground states that challenge theory2. The emergence of transition metal dichalcogenide moiré superlattices provides a highly controllable platform in which to study long-range electronic correlations5–12. Here we report an observation of nearly two dozen correlated insulating states at fractional fillings of tungsten diselenide/tungsten disulfide moiré superlattices. This finding is enabled by a new optical sensing technique that is based on the sensitivity to the dielectric environment of the exciton excited states in a single-layer semiconductor of tungsten diselenide. The cascade of insulating states shows an energy ordering that is nearly symmetric about a filling factor of half a particle per superlattice site. We propose a series of charge-ordered states at commensurate filling fractions that range from generalized Wigner crystals7 to charge density waves. Our study lays the groundwork for using moiré superlattices to simulate a wealth of quantum many-body problems that are described by the two-dimensional extended Hubbard model3,13,14 or spin models with long-range charge–charge and exchange interactions15,16. © 2020, The Author(s), under exclusive licence to Springer Nature Limited.