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. The cornerstone of this program is a ``z-adic'' variant of Grothendieck's l-adic monodromy theorem. \\ \\ The talk is aimed at a general audience of number theorists and arithmetic geometers. No special knowledge of monodromy theory or Drinfeld modules is assumed.