  We prove several results concerning the existence of potentially crystalline
lifts with prescribed Hodge-Tate weights and inertial types of a given
n-dimensional mod p representation of the absolute Galois group of K, where
K/Q_p is a finite extension. Some of these results are proved by purely local
methods, and are expected to be useful in the application of automorphy lifting
theorems. The proofs of the other results are global, making use of automorphy
lifting theorems.
