In this series of talks I will try to give an overview of the main issues that arise when trying to establish mod p and p-adic analogues of the theorems of Langlands, Deligne, Drinfeld and Carayol, describing the l-adic (l ≠ p) étale cohomology of the Drinfeld and Lubin-Tate towers. The methods are very different, based on perfectoid spaces, representation theory of GL2(Qp) and its inner form, as well as syntomic cohomology and the six-functor formalism due to Lucas Mann. This is based on joint work with Colmez and Niziol (for the Drinfeld tower) and Camargo (for the Lubin-Tate tower).