Two basic questions in the study of Galois representations are whether a mod $p$ representation of the Galois group of a number field lifts to a $p$-adic representation, and the much subtler question of whether it moreover lifts to a geometric $p$-adic representation. I will survey some key examples of these problems—both known and unknown aspects—and describe some recent joint work with N. Fakhruddin and C. Khare on lifting irreducible mod p representations valued in general reductive groups.