I review the theory side of the synergetic international effort of experimentalists and theorists in Compton scattering on one- and few-nucleon systems. It is an excellent opportunity to probe the symmetries and strengths of nucleonic and nuclear interactions and relate to lattice-QCD computations of fundamental hadronic properties. Rich information is encoded in the polarisabilities, which parametrise the stiffness of charge distributions against deformations. The spin polarisabilities are particularly interesting since they parametrise the stiffness of the spin in external electro-magnetic fields (nucleonic Faraday effect) and probe the spin-dependent component of the pion-nucleon interaction. I then discuss tests and extractions of the polarisabilities of the proton and neutron from data. Comprehensive studies show how sensitive observables for nuclei up to ${}^4$He are on the individual nucleonic scalar and spin polarisabilities, and to their combinations. We are also developing statistical methods to identify experiments with the likely biggest impact on extracting values from future data. This facilitates planning and analysis of the new generation of Compton experiments.