The classification of crystalline representations has played a key role in the modularity lifting theorems of the type first proved by Wiles and Taylor-Wiles, and then extended by various people. I will explain a new classification of such representations which works in the presence of ramification.

Its applications include a classification of p-divisible groups and finite flat group schemes conjectured by Breuil, and a conjecture of Fontaine which asserts that a crystalline representation of Hodge-Tate weights 0,1 comes from a p-divisible group.