Conformal partial wave expansion provide Fourier-like decompositions of correlation functions in Conformal Field Theory. Despite their fundamental importance, conformal partial waves remain poorly understood, at least beyond the case of four local fields. In the last few years, a deep relation with integrable quantum mechanical models has emerged. It offers a wealth of powerful new algebraic methods to study and construct conformal partial waves e.g. for general supermultiplets, non-local (line-, surface-) operators and multi-point correlation functions. In my talk I will use ideas from harmonic analysis of the conformal group to embed conformal partial waves into the framework of Gaudin integrable models and then discuss several concrete ramifications as trigonometric and elliptic Calogero-Sutherland models. The latter are relevant for multi-point blocks of scalar fields. ----- Part of London Integrability Journal Club. Please register at integrability-london.weebly.com/registration.html if you are a new participant.