Kurzbeschreibung: In this talk we give a short overview about the theory of harmonic weak Maass forms and their applications in number theory and (arithmetic) geometry. First, we will briefly describe how the study of generating series of number theoretic functions is related to modular forms. Then we will introduce harmonic weak Maass forms, real-analytic generalizations of modular forms. We will describe how the vanishing of the central L-derivative of rational elliptic curves is related to the properties of the Fourier coefficients of certain harmonic weak Maass forms and indicate how this result can be generalized. (In parts this is joint work with Jan Bruinier and Markus Schwagenscheidt.)