Conformal vector fields in general relativity

A general discussion of conformal vector fields in space-times is given. Amongst the topics considered are the maximum dimension of the conformal algebra for space-times that are not conformally flat, the nature of conformal isotropies and a new approach to the theorem of Bilyalov and Defrise-Carter concerning the reduction of the conformal algebra to a Killing or homothetic algebra. Some deficiencies in the original statements of this theorem are discussed (with reference to a general class of counterexamples) and corrected. The proof offered is geometrical in nature and has the advantage of displaying some of the more general features and properties of conformal vector fields and the ways in which they can differ from Killing vector fields.