As the title indicates, this talk will be about classical, function-theoretic harmonic analysis. However, I will discuss some recently discovered structures in harmonic analysis which may have sheaf-theoretic analogs. More precisely, I will discuss the theory of asymptotics on spherical varieties as it appears in the work of Bezrukavnikov-Kazhdan (for the group) and my joint work with Venkatesh (in general), as well as the scattering theory which allows to formulate a Paley-Wiener theorem (joint work with Delorme and Harinck). There will be some overlap with my talk at the NGM Research Seminar at MSRI in October, but the emphasis will be different.