Until quite recently, many points of connection between mathematics and theoretical physics focused on so-called "twists" of supersymmetric field theories. Such twists are holomorphic or topological field theories that are amenable to rigorous mathematical descriptions; intuitions from supersymmetric field theory then provide surprising insights into relations between twisted theories, mirror symmetry being a prime example. I will discuss some recent work, building on a powerful analogy between supergeometry and almost-complex geometry, that allows one to formulate a theory and all of its twists in uniform fashion. This gives rise to a more mathematically natural, intrinsically geometric, and computationally tractable characterization of supersymmetric field theories and supergravity theories. As an example, I will discuss the theory of eleven-dimensional supergravity, showing that it is structurally identical to the moduli problem of deformations of a Calabi-Yau twofold. I will go on to discuss some concrete progress towards constructing the long-conjectural (2,0) superconformal field theories in six dimensions, making use of similar techniques. Surprisingly, several of the exceptional simple infinite-dimensional Lie superalgebras found by Kac appear in central roles.