Conformal blocks for correlation functions of spinning operators play an increasingly important role for the conformal bootstrap programme. In my talk, after a brief overview of the conformal bootstrap programme in d>2, I will present our approach to such spinning blocks through the harmonic analysis of certain bundles over a double-coset of the conformal group. This approach furnishes solutions of Casimir equations for external fields of arbitrary spin and dimension in terms of functions on the conformal group. This allows us to reinterpret standard operations on conformal blocks in terms of group-theoretic objects. As an application of these ideas, I will identify the set of seed blocks in any dimension. On the other hand, the resulting Casimir equations are given by a matrix version of the Calogero-Sutherland Hamiltonian what connects the theory of conformal blocks with the rich world of integrable quantum systems. At the end of my talk, I will discuss the generalisation of these ideas to CFT with boundaries and defects as well as supersymmetric CFTs. Slides can be found here.