We study the sheaf of Kähler differentials on the arc space of an algebraic variety. We obtain explicit formulas that can be used effectively to understand the local structure of the arc space. The approach leads to new results as well as simpler and more direct proofs of some of the fundamental theorems in the literature. The main applications include: an interpretation of Mather discrepancies as embedding dimensions of certain points in the arc space, a new proof of a version of the birational transformation rule in motivic integration, a new proof of the curve selection lemma for arc spaces, and a description of Nash blow-ups of jet schemes. This is joint work with Tommaso de Fernex.