A major part of the Langlands programme is a conjectural relationship between number-theoretic objects, such as Galois representations, and analytic ones, such as automorphic forms.  These conjectures are now mostly proved in the context of classical modular forms on the one hand, and two-dimensional representations of Galois groups over Q on the other.  One of the key results is Serre's Conjecture, proved by Khare and Wintenberger, which can be viewed as a mod p Langlands correspondence.  An important feature of Langlands correspondences is a local-global compatibility principle, but it is not even known how to formulate a conjectural mod p version of this principle beyond the context of classical modular forms.  I'll discuss what's known and what some of the difficulties are.