The discovery of fractional quantum anomalous Hall (FQAH) states in moiré materials has opened the possibility of realizing phases of dispersing and interacting anyons. In this talk, I will show how anyon dispersion and binding arise microscopically in FQAH states and lead to striking experimental consequences. I will first argue that the interplay of anyon dispersion and disorder-induced localization provides a natural explanation for the recent observation of superconductivity and re-entrant integer quantum Hall states upon doping the 2/3 FQAH state in twisted MoTe₂: doping anyons drives plateau transitions of composite fermions into integer quantum Hall states, realizing either a superconducting phase or a re-entrant integer quantum Hall phase.

Going beyond long-distance effective theory, I will then introduce a controlled and scalable approach, based on projection onto the space of Laughlin quasiholes, that captures anyon energetics at short distances and enables accurate computation of both single-particle dispersion and multi-anyon binding. Applied to twisted MoTe₂, the theory predicts a quasihole bandwidth of order 1 meV, indicating that itinerant-anyon physics should be experimentally relevant. Applied to Laughlin quasiholes with screened Coulomb interactions, it predicts bound-state formation when the screening length is comparable to or smaller than the magnetic length — remarkably, despite both the bare electron-electron interaction and the quasihole electrostatic potential being purely repulsive. The binding is a Berry-phase effect, driven by oscillations in the quasihole density profile invisible at the classical level. I will map out a sequence of phases — free e/3 anyons, paired 2e/3 bound states, three-anyon charge-e clusters, and larger composites — and discuss implications for charge-imaging experiments and the phase diagram of itinerant anyons in FQAH materials.