The arc space of an algebraic variety and its connection to singularities have been studied since the work of Nash in the 1960s. While classically this involved only the topology of the arc space, recently there has been progress describing the schematic structure and relating it to invariants of singularities. In this talk I will present some results in this direction. One of the main ingredients here is de Fernex-Docampo’s description of the sheaf of differentials on the arc space, and I will hint at a possible generalization of their result to other contexts. This talk covers joint work with Tommaso de Fernex and Roi Docampo.