On the reduction of Galois representations

I will describe the problem of mod p reduction of p-adic Galois representations.  For crystalline representations, the reduction can be computed using the compatibility of p-adic and mod p Local Langlands Correspondences; this method was first introduced by Breuil in 2003.  After giving a brief sketch of the history of the problem, I will discuss how the reductions behave for representations with slopes in the half-open interval [1,2).  This is based on joint works with Eknath Ghate, and also with Sandra Rozensztajn for slope 1.