  The present review includes the description of theoretical methods for the
investigations of the spectra of hydrogen-like systems. Various versions of the
quasipotential approach and the method of the effective Dirac equation are
considered. The new methods, which have been developed in the eighties, are
described. These are the method for the investigation of the spectra by means
of the quasipotential equation with the relativistic reduced mass and the
method for a selection of the logarithmic corrections by means of the
renormalization group equation. The special attention is given to the
construction of a perturbation theory and the selection of graphs, whereof the
contributions of different orders of $\alpha$, the fine structure constant, to
the energy of the fine and hyperfine splitting in a positronium, a muonium and
a hydrogen atom could be calculated.
  In the second part of this article the comparison of the experimental results
and the theoretical results concerning the wide range of topics is produced.
They are the fine and hyperfine splitting in the hydrogenic systems, the Lamb
shift and the anomalous magnetic moments of an electron and a muon. Also, the
problem of the precision determination of a numerical value of the fine
structure constant, connected with the above topics, is discussed.
