The notion of "causal structure" has been used in General Relativity most of the time in connection with "conformal structure" or "conformal equivalence". Although these concepts of causal structure have proven themselves quite useful, there are cases in which two spacetimes have similar causal properties but no conformal mapping between each other exists (e. g. two weak perturbations of Minkowski spacetime). To remedy this, the concept of "causal map" is introduced.
Causal maps generalize conformal maps and we show how they can be used to define a new concept of causal structure which contains the traditional one based on conformal equivalence as a particular case. Causal structures can be sorted by means of a partial order and we show the extent to which this ordering generalises the classification of spacetimes according to the "standard hierarchy of causality conditions". We also put forward the concept of stability and instability of the causal structure and explain how it can be applied to examples as relevant as Minkowski or de Sitter spacetimes.
The causal structure of other well-known spacetimes (Schwarzschild, pp-waves) is also studied with our techniques.