There are many ways of defining persistence diagrams. In this talk I will discuss the definition based on the Mobius inversion function which was introduced by Amit Patel under the name Generalized Persistence Diagrams. I will cover how this approach has appeared implicitly and explicitly in various results on persistence as well as various implications of this approach and (very) new developments. In particular, I will cover a surprising connection between Euler characteristics and persistence diagrams.  This is joint work with Amit Patel.