  This PhD thesis is devoted to the low-energy structure of the nucleon (proton
and neutron) as seen through electromagnetic probes, e.g., electron and Compton
scattering. The research presented here is based primarily on dispersion theory
and chiral effective-field theory. The main motivation is the recent proton
radius puzzle, which is the discrepancy between the classic proton charge
radius determinations (based on electron-proton scattering and normal hydrogen
spectroscopy) and the highly precise extraction based on first muonic-hydrogen
experiments by the CREMA Collaboration. The precision of muonic-hydrogen
experiments is presently limited by the knowledge of proton structure effects
beyond the charge radius. A major part of this thesis is devoted to calculating
these effects using everything we know about the nucleon electromagnetic
structure from both theory and experiment.
  The thesis consists of eight chapters. The first and last are, respectively,
the introduction and conclusion. The remainder of this thesis can roughly be
divided into the following three topics: finite-size effects in hydrogen-like
atoms, real and virtual Compton scattering, and two-photon-exchange effects.
