The present status and recent developments in the theory of light hydrogenic atoms, electronic and muonic, are extensively reviewed. The discussion is based on the quantum field theoretical approach to loosely bound composite systems. The basics of the quantum field theoretical approach, which provide the framework needed for a systematic derivation of all higher order corrections to the energy levels, are briefly discussed. The main physical ideas behind the derivation of all binding, recoil, radiative, radiative-recoil, and nonelectromagnetic spin-dependent and spin-independent corrections to energy levels of hydrogenic atoms are discussed and, wherever possible, the fundamental elements of the derivations of these corrections are provided. The emphasis is on new theoretical results which were not available in earlier reviews. An up-to-date set of all theoretical contributions to the energy levels is contained in the paper. The status of modern theory is tested by comparing the theoretical results for the energy levels with the most precise experimental results for the Lamb shifts and gross structure intervals in hydrogen, deuterium, and helium ion He + , and with the experimental data on the hyperfine splitting in muonium, hydrogen and deuterium. *

Corrections to hyperfine splitting and Lamb shift of order α 2 (Zα) 5 induced by the diagrams with radiative photon insertions in the electron line are calculated in the Fried-Yennie gauge. These contributions are as large as −7.725(3)α 2 (Zα) 5 /(πn 3 )(m r /m) 3 m and −0.6711(7)α 2 (Zα)/(πn 3 )E F for the Lamb shift and hyperfine splitting, respectively. Phenomenological implications of these results are discussed with special emphasis on the accuracy of the theoretical predictions for the Lamb shift and experimental determination of the Rydberg constant. New more precise value of the Rydberg constant is obtained on the basis of the improved theory and experimental data.

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The complete contribution of diagrams with the light-by-light scattering to the Lamb shift is found for muonic hydrogen, deuterium and helium ion. The results are obtained in the static muon approximation and a part of the paper is devoted to the verification of this approximation and analysis of its uncertainty.

The results and main steps of an analytic calculation of radiative-recoil corrections of order α(Zα) 5 (m/M )m to the Lamb shift in hydrogen are presented. The calculations are performed in the infrared safe Yennie gauge. The discrepancy between two previous numerical calculations of these corrections existing in the literature is resolved. Our new result eliminates the largest source of the theoretical uncertainty in the magnitude of the deuterium-hydrogen isotope shift.

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