What is a twist of a supersymmetric field theory?

Twists of supersymmetric field theories are a rich source of theories which depend topologically and/or holomorphically on different directions of spacetime. A key characteristic of these theories is that they can be placed on curved manifolds on which their untwisted ancestors could not have been defined. In certain cases, one then recovers existing manifold invariants (e.g. Donaldson invariants) and in others, produces new ones (e.g. Gromov-Witten invariants). Moreover, the twisting procedure critically preserves a chosen supersymmetry, which means that often, the twisted theory is a powerful tool for studying aspects of the untwisted theory. I will attempt to give an overview of how all of this works, giving examples along the way.