Algebraic geometry has allowed geometry to reach more abstract objects in more general categories. An example of this is noncommutative geometry. In this theory, no notion of point can be defined. A nice middle step towards noncommutative manifolds (and other abstract geometric objects in modern algebraic geometry) is the category of supermanifolds. These manifolds are already out of reach of classical geometry (Riemannian, differential, etc.), and the usual notion of point is not good enough. In the talk, I will start with a fast introduction to supergeometry following two of the standard approaches, the second one widely known and used by the community. My goal is to present a modified version of the so-called functor of points of a supermanifold, which allows to enlarge the category of supermanifolds and allows to give a more intuitive notion of point in such a geometric ambience which, restricted to the classical geometry case, coincides with the usual. For this, it will be necessary to introduce some ideas from category theory, which are quite important for any theory concerning algebraic geometry.