Abstract: In this talk I report on progress made in understanding the concept of a Cayley map for algebraic supergroups. First I introduce the concept of the classical Cayley map between a connected algebraic group and its Lie algebra and discuss two kinds of generalization. As a second step I introduce some concepts of so-called 'super-mathematics'. The two ingredients are then combined to get a classical Cayley map for the supergroup OSp(n|2m).