While the highest weight representation theory of affine Lie algebras in characteristic zero is by now fairly well understood, far less has been known in characteristic p > 0. We will report on some basic results in this direction, e.g. the Harish--Chandra center of the enveloping algebra at all levels, the linkage principle, an analogue of the Kac--Kazhdan conjecture, and a local de Rham geometric Langlands correspondence. This is based on joint work with Ivan Losev.