This chapter presents a general analysis of Kodaira and Legendre moduli spaces of compact submanifolds which shows that the phenomenon of inducing rich geometric structures on these creatures is a very general one. In twistor theory one encounters a law of nature rather than a fortunate concurrence of circumstances. As a by-product, the chapter shows that several twistor constructions follow from a couple of general theorems which say that moduli spaces of compact complex submanifolds often come equipped with families of torsion-free connections satisfying some natural integrability conditions. A surprising feature of this theory is how easily one can estimate the holonomy group of induced connections or check non-triviality of the resulting geometry.