The arc space of an algebraic variety parametrizes formal germs of smooth curves mapping into the variety. It is an infinite dimensional scheme whose geometry can be used to analyze and control invariants of singularities. Traditionally this is done by focusing on the topological structure of arc spaces, but recent developments suggest that its scheme structure (its singularities, its non-reduced scheme structure) should also play an important role. For instance, embedding dimensions and codimensions in local rings of arc spaces are related to the computation of discrepancies and to a theorem of Drinfeld, Grinberg, and Kazhdan (DGK) on formal neighborhood of arcs. In collaboration with C. Chiu and T. de Fernex, we have been developing a toolbox for the study of the scheme structure of arc spaces and jet schemes. The starting point is an explicit formula for the sheaf of differentials on the arc space, which leads to new results (a converse to the DGK theorem, the control of arc fibers, and a description of Nash blow-ups of jet schemes) as well as simpler and more direct proofs of some of the fundamental theorems in the literature (numerical versions of the birational transformation rule in motivic integration and the Denef-Loeser lemma, and a new proof of the curve selection lemma for arc spaces). In this talk, I will give an overview of the latest developments in this area.