The aim of this talk is to give a (partial) description of the set of functions that are both positive and positive definite (that is, with a positive Fourier transform): in short PPDs. It turns out that this set has a rather rich structure for which a full description seems out of reach. Further, the work establishes a link between the question of describing the set of PPDs and the question of having simple criteria to check whether a function is positive definite, without computing its Fourier transform. \ \ (Lecture in Mathematics For Physicists)