© 2010, Birkhäuser, Springer Basel AG. This chapter is a brief summary of the main concepts and results about pseudo-Riemannian and Lorentzian manifolds which will be widely used in the rest of the book. While the theory is presented from the beginning in a systematic way, some proofs have been omitted and others are simply hinted at. That is why the reading of this chapter assumes previous knowledge of the subject. We refer to  for further treatment.