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.