Precise physics of simple atoms

The simplicity of "simple" atoms has been for a while a challenge to precision theory and experiment. Are the hydrogen-like atomic systems simple enough to be calculated with an accuracy, appropriate to compete with the best experimental results? That is a question, that theorists have tried to answer. The simplest atoms are different two-body bound systems with a low value of the nuclear charge: Z = 1 (hydrogen, deuterium, muonium and positronium) and Z = 2 (ions of helium-3 and helium-4) etc, We do not try to review theoretical calculations (if necessary details can he found in Ref, [1]), but present state of art in physics of simple atoms and discuss in detail the theoretical and experimental status of studying such atoms.