In quantum field theory, "twisting" is a procedure for producing new theories from old, where the new theories have particularly nice symmetry properties (for example, topological quantum field theories). The twisting construction involves the choice of a nilpotent element of a super Lie algebra that acts on the theory. I will discuss joint work with Owen Gwilliam and Matteo Lotito on twisting for theories with an action of a superconformal algebra and the appearance of interesting algebraic structures such as vertex algebras and $E_n$ algebras.