  Quantum electrodynamics is the first successful and still the most successful
quantum field theory. Simple atoms, being essentially QED systems, allow highly
accurate theoretical predictions. Because of their simple spectra, such atoms
have been also efficiently studied experimentally frequently offering the most
precisely measured quantities. Our review is devoted to comparison of theory
and experiment in the field of precision physics of light simple atoms. In
particular, we consider the Lamb shift in the hydrogen atom, the hyperfine
structure in hydrogen, deuterium, helium-3 ion, muonium and positronium, as
well as a number of other transitions in positronium. Additionally to a
spectrum of unperturbed atoms, we consider annihilation decay of positronium
and the g factor of bound particles in various two-body atoms. Special
attention is paid to the uncertainty of the QED calculations due to the
uncalculated higher-order corrections and effects of the nuclear structure. We
also discuss applications of simple atoms to determination of several
fundamental constants.
