TThis research project builds on two previous (still running) projects financed by FWO and BOF, respectively. The central notion of the project is that of a Lie incidence geometry; such geometries are the natural geometries of the simple groups of Lie type, including the isotropic simple algebraic groups, and all simple classical groups. We use the Lie incidence geometries to find and prove properties of the corresponding groups, often inspired by the Freudenthal-Tits Magic Square (which is also a source of interesting problems for us). We concentrate on the behaviour of the groups with respect to the graph distance between the points of the geometries, in particular "opposition" (maximal distance). We also envisage to investigate the finite case by defining analogues of blocking sets of projective spaces  in Lie incidence geometries in different ways. These should give rise to drastic generalisations of fundamental theorems like the one of Bose & Burton (1966).