This monograph offers a comprehensive summary of experiments on Compton scattering from the proton and neutron and 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, and their analysis and results. Investigating the same reaction on the neutron requires a focus on the deuteron as a neutron target. Based on this success, the experiment on the bound neutron was developed. The publication of this series of Compton scattering experiments enables investigators ready access to these contributions to our understanding of the electromagnetics as well as the backward spin polarizabilities of the proton and neutron.