Abstract: A Lagrangian inclusion is a smooth map from a compact real surface into $C^2$ which is a local Lagrangian embedding except a finite set of singular points. The singular points can be taken to be either transverse double self-intersection points or the so-called open Whitney umbrellas. In the first talk I introduce relevant terminology and will formulate a recent result (joint with A. Sukhov) concerning rational convexity of a Lagrangian inclusion with one umbrella point. As an application I will explain how Lagrangian surgery can be used to obtain some approximation results on real surfaces.