Non-linear physical systems and their mathematical structure form one of the most active fields in present mathematics and mathematical physics. This volume covers parts of that topic. It reports on differential geometrical and topological properties of those non-linear systems, which can be viewed physically as models for quantized non-relativistic particles constrained, i.e. localized, on a (smooth) manifold or as classical or quantized fields with non-linear field equations. The contributions of this volume show how to deal with these different types of non-linearities. There are various physically motivated approaches to both of them. For systems constrained on a manifold generically geometric methods are used with promising mathematical and physical results. Now that the feeling has dissipated, that global solutions of non-linear field equations are"extra - terrestrial beasts" (see the contribution of I.E. SEGAL), also here a more global and geometrical approach is applied with extreme success, we refer e.g. to the application of twistor geometry or to the analysis of solution manifolds of non-linear equations. The structures of both types of non-linearities are deeply related.