In this chapter we shall embark upon an axiomatization of Einstein-Weyl causality on a set M consisting of points in the sense of Euclidean geometry. Our aim is to investigate the mathematical consequences of Einstein-Weyl causality at the local level, and we therefore assume that M carries no prede fined mathematical structure. Causal structures have been investigated by other authors, but mainly on spaces which already carried other structures such as a manifold structure (the work of Kronheimer and Penrose is a notable exception). Two of these will be discussed in some detail in Chap. 10.