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

Nucleon Polarizabilities: from Compton Scattering to Hydrogen Atom