We give an overview of low-energy Compton scattering (gamma^(*) p --> gamma p) with a real or virtual incoming photon. These processes allow the investigation of one of the fundamental properties of the nucleon, i.e. how its internal structure deform
s under an applied static electromagnetic field. Our knowledge of nucleon polarisabilities and their generalization to non-zero four-momentum transfer will be reviewed, including the presently ongoing experiments and future perspectives.
