Abstract: The recent observation of fractional Chern insulators—lattice analogs of fractional quantum Hall phases realized in the absence of an external magnetic field—in two‑dimensional moiré materials opens a new avenue for the study and control of anyons, their elementary quasiparticle excitations. These platforms offer two qualitatively new opportunities: stabilizing itinerant phases of anyons and deterministically trapping individual anyons. First, I will discuss the dynamics of anyons in the absence of a magnetic field. In conventional fractional quantum Hall systems, the kinetic energy of electrically charged anyons is quenched, and they tend to localize at defects or impurities. In fractional Chern insulators, by contrast, anyons can acquire a dispersion and form itinerant phases. I present a concrete example in which a non‑uniform Berry curvature generates a nontrivial band structure for anyons, leading to dispersive anyonic excitations. If time allows, I will briefly outline possible emergent phases of mobile anyons, including the prospect of anyon superconductivity in a concrete microscopic model. Finally, I will demonstrate how gate engineering in moiré materials enables the controlled trapping of anyons at user‑defined locations. Using a microscopic model, I show that anyons can be localized away from defects or impurities, providing a route toward the programmable manipulation of anyons in solid‑state platforms.