The talk will present the concept of the Maximal Equicontinuous Factor of a flow (see Auslander: Minimal Flows and their Extensions (1988)). It will be shown that this factor has monotonicity properties, i.e. that preimages of singletons are connected (new result by T.Hauser and T.Jäger). Furthermore, as an application of this result, a characterization of minimal toral homeomorphisms will be given (new result by T.H. and T.J.).