Diagram geometries and chamber systems of various types have been used and investigated intensively in recent years - not only in finite group theory, but in other areas as well. This development has led to a need for some clarification of the variations and generalizations introduced by the many authors, and for a discussion of the different axiomatic approaches. The authors concentrate on the approach based on incidence geometry and discuss many important local and global properties of interest such as transversality, all sorts of connectedness and aspects of introducing types on the geometry. Besides explicit statements of the various interrelations many examples are given,hence the reader is provided with a lot of detailed helpful information.