A development of twistor theory, 4-dimensional conformal field theory, has led to an interest among twistor theorists in conformally flat manifolds, since their twistor spaces play the role in 4-dimensional CFT that Riemann surfaces take on in 2-dimensional CFT. This chapter describes a period mapping, analogous to the period mapping for Riemann surfaces, which may prove to be important in the study of conformally flat 4-manifolds. The chapter introduces the condition of ‘strong regularity’ for conformal structures on a 4-manifold and show that period mapping is much simpler for such structures. The second part of the chapter shows the existence of interesting strongly regular conformally flat manifolds; this is the intended point of departure for a more detailed study of period mapping.