This monograph offers a comprehensive summary of experiments on Compton scattering from the proton and neutron performed at the electron accelerator MAMI.  The experiments cover a photon energy range from 30 MeV to 500 MeV. The reader is introduced to the theoretical concepts of Compton scattering, followed by a description of the experiments on the proton, their analysis and results.  Investigating the same reaction on the neutron requires a focus on the deuteron as a neutron target. Based on the success of the first quasi-free Compton scattering experiment on the bound proton, the experiment on the bound neutron was developed.  The publication of these series of Compton scattering experiments enables investigators to improve our understanding of the electromagnetic and the spin polarizabilities of the proton and neutron.