In the past decade, $p$-adic Hodge theory has been transformed by the discovery of perfectoid spaces and the Fargues–Fontaine curve. One area, however, that has remained almost untouched by these breakthroughs is the theory of $p$-adic differential equations, as initiated by Dwork and Robba. After all, perfectoid spaces do not carry interesting differential forms in the naive sense. In this talk, I will explain how to address this issue and, more generally, how one can define $\mathcal{D}$-modules on arc-stacks living over $\mathbb{Q}_p$. I will then focus on $\mathcal{D}$-modules on the Fargues–Fontaine curve and sketch a new proof of the cornerstone of the theory of $p$-adic differential equations, namely the $p$-adic monodromy theorem. Finally, I will outline how this perspective also leads to new results and conjectures on $p$-adic cohomology theories, such as Hyodo–Kato cohomology. Based on joint work with Anschütz, Le Bras, Rodriguez Camargo, and Scholze.