Lorentzian geometry is an active field of mathematical research with particular relations to relativity theory.The main purpose of this exposition is to present novel results in a concise overview and to provide a kind of mind map of the research done by the community today. We hope in particular that this little book serves young researchers who are interested in this topic as a valuable source for information on the frontiers of this field. It includes articles and surveys covering a large variety of topics in Lorentzian geometry, amongst which are:. Submanifolds of particular causal character, . Killing and conformal vector fields, . warped products and foliations, . Einstein equations, energy conditions and black holes, . techniques from Riemannian geometry, . algebraic classification of spacetimes.