This will be a gentle introduction to some basic techniques developed 15 years ago for lifting two-dimensional Galois representations from characteristic $p$ to characteristic zero. The talk will include motivation as to why one would want such a construction and applications of the method.

In the second part I will talk about recent joint work with Khare that shows one can lift a two-dimensional mod $p$ representation to an $O$-valued representation where $O$ is any ramified extension of the $p$-adic integers.