After Plancherel formula

We discuss two topics related to Fourier transforms on Lie groups and on homogeneous spaces: the operational calculus and the Gelfand–Gindikin problem (program) about separation of non-uniform spectra. Our purpose is to indicate some non-solved problems of non-commutative harmonic analysis that definitely are solvable. This is a sketch of my talks on VI School “Geometry and Physics”, Bialowieza, Poland, June 2017.