Abstract: Recent experimental breakthroughs on hetero-structures composed of twisted atomically thin semiconducting layers have demonstrated (i) the existence of phases analog to fractional quantum Hall states in the absence of magnetic fields, and (ii) a high degree of tunability and versatility in sample design, including non-uniform gating or metallization-induced superconductivity. In this talk, I will review these achievements and argue why, together, they promote moiré systems as unique platforms to realize, manipulate and control anyons. These long sought-after quasiparticles possess, amongst other properties, fractional statistics and can be harnessed to perform fault-tolerant quantum computation. For this, I will use concrete models that integrate the aforementioned experimental building blocks and can yield, if realized, localized anyonic quasi-particles never isolated before or novel itinerant anyonic phases that could not exist in quantum Hall systems.