Moiré materials have emerged in the last years as a highly tunable and experimentally accessible platform to study strongly correlated topological phases of matter. In particular, twisted bilayers of graphene and transition metal dichalcogenides have been shown to be Chern insulators, where electrons break time-reversal symmetry and spontaneously polarize into a single spin and valley, occupying a band with non-zero Chern number. While Chern insulators are already interesting because they exhibit the integer quantum Hall effect in the absence of an external magnetic field, recent theoretical predictions and experimental realizations of fractional Chern insulators (FCIs) have opened up the possibility of exploring even more exotic states. These include phases hosting non-Abelian excitations, which are interesting for fault-tolerant topological quantum computing, states with Chern number higher than one that have no analog in the standard quantum Hall paradigm, and quantum Hall crystals exhibiting topological order with broken translation symmetry. In recent work, we have predicted the emergence of these exotic phases and provided insights into the underlying mechanisms. Now, we aim to extend the landscape of many-body phases to the realm of bosonic topological order based on moiré excitons. In this project we will use memory-expensive numerical methods to investigate bosonic topological order based on long-lived moiré excitons that are accessible in experiments. After setting up the many-body Hamiltonian projected into the relevant flat bands, we will employ exact diagonalization to obtain the energetically lowest eigenstates of the system. The initial identification of the system’s phase will be based on the ground state degeneracy and the respective (momentum) quantum numbers. A more complete analysis will include the calculation of many-body Chern number (yielding the observable quantization of Hall resistance) and particle entanglement spectra (which contain information about the phase-specific quasi-hole excitations). In addition, the structure factor and pair-correlation function can be computed to explore the liquid or crystal character of the ground state. Using these methods, we aim to shed light on the stability and nature of competing phases such as charge density waves, bosonic Hall crystals, superfluids, and Abelian as well as non-Abelian exciton FCIs. We expect our results to provide important insights on fundamental topics of condensed matter quantum many-body physics with potential impact in the future realization of topologically-protected quantum devices.