  We review the current state of knowledge of the nucleon polarizabilities and
of their role in nucleon Compton scattering and in hydrogen spectrum. We
discuss the basic concepts, the recent lattice QCD calculations and advances in
chiral effective-field theory. On the experimental side, we review the ongoing
programs aimed to measure the nucleon (scalar and spin) polarizabilities via
the Compton scattering processes, with real and virtual photons. A great part
of the review is devoted to the general constraints based on unitarity,
causality, discrete and continuous symmetries, which result in
model-independent relations involving nucleon polarizabilities. We (re-)derive
a variety of such relations and discuss their empirical value. The proton
polarizability effects are presently the major sources of uncertainty in the
assessment of the muonic hydrogen Lamb shift and hyperfine structure. Recent
calculations of these effects are reviewed here in the context of the
"proton-radius puzzle". We conclude with summary plots of the recent results
and prospects for the near-future work.
