The recursive realizability interpretation of constructive logic, first developed by S.C. Kleene and encoded in topos-theoretic form in the late 1970s by Martin Hyland, is only one of a large range of weird and wonderful toposes which can be constructed in the same way from simple algebraic structures known as Schonfinkel algebras (or partial combinatory algebras). In this talk I'll describe the construction and indicate something of the range of different examples available; subsequent talks will be devoted to investigating how they fit together into a 2-category.