Modularity lifting is a deep and technical methodology introduced by Wiles and Taylor-Wiles in the proof of Fermat's Last Theorem. Modularity lifting theorems have played a central role in many subsequent developments, including the proofs of the Sato-Tate conjecture for elliptic curves over totally real fields and Serre's conjecture.  We explain why modularity lifting theorems are important, describe what the statement of such a theorem roughly looks like, and sketch the several types of arguments that are involved in their proof.