The theory of Drinfeld modules is remarkably similar to the theory of abelian varieties, but their local monodromy behaves differently and is poorly understood. In this talk I will present a research program which aims to fully describe this monodromy. To facilitate the understanding I will also give an overview of the classical monodromy theory for algebraic varieties. The talk is aimed at a general audience of arithmetic geometers and needs no prerequisites apart from the definition of étale cohomology. In particular no preliminary knowledge of Drinfeld modules is required.