Abstract: Jet spaces, also known as jet bundles or higher-order phase spaces, encode infinitesimal information about curves mapping to a manifold or other geometric objects. They appear naturally in differential geometry and topology, and they provide a geometric context to deal with higher derivatives, Taylor expansions, and ordinary differential equations. In algebraic geometry, jets and jet spaces are used to define and control invariants of algebraic varieties, and play a prominent role in singularity theory. Thinking of p-adic integers as analogous to power series, one is led to the notion of arithmetic jet space. We replace a manifold with a system of Diophantine equations, and the corresponding arithmetic jet space encodes p-adic solutions. Derivations are replaced by what are known as p-derivations, a notion closely related to the Frobenius map. Arithmetic jet spaces appear naturally in arithmetic geometry, particularly when trying to understand orbit spaces of group actions. They also play a important role in p-adic Hodge theory. In collaboration with Tommaso de Fernex and Christopher Chiu, we are studying differential forms on arithmetic jet spaces. This is motivated by previous work, where we described modules of differentials on geometric jet spaces and explained how they can be used to effectively control invariants of singularities. In the arithmetic context we get very similar descriptions, which we expect to use in the future to relate singularity invariants across different characteristics. Our proofs rely heavily on category theory. In fact, our formulas for modules of differentials on jet spaces are obtained as corollaries to a very general theorem about adjunctions and tangent categories, which is of independent interest in category theory.