On lifting of embeddings between transitive models of set theory

Suppose M and N are transitive models of set theory, P is a forcing notion in M and G is P-generic over M. An elementary embedding j : (M, ∈) → (N, ∈) lifts to M[G] if there is j+ : (M[G], G, ∈) → (N[j+(G)], j+(G), ∈) such that j+ restricted to M is equal to j. We survey some basic applications of the lifting method for both large cardinals and small cardinals (such as ω2, or successor cardinals in general). We focus on results and techniques which appeared after Cummings’s handbook article [Cum10]: we for instance discuss a generalization of the surgery argument, liftings based on fusion, and compactness principles such as the tree property and stationary reflection at successor cardinals.