The results of theoretical and experimental studies of the Stark broadening of hydrogen lines in plasmas made during the last 10--15 years are reviewed. Plasmas and the hydrogen atom are the objects in which specific features of the Coulomb interaction manifest themselves most clearly. The statistics of the Coulomb microfield produced by the ions and electrons in a plasma (with allowance for the effects of screening and ion-ion correlations) and the temporal fluctuations of that microfield are discussed briefly. The dynamics of the interaction of the plasma microfield with the radiating atom is analyzed in detail. In addition to discussing the well known approximations in the theory of line broadening--the impact and statistical approximations--considerable attention is given to the intermediate spectral region in which both those approximations break down. The possibility of describing this region is due to a specific (fourdimensional) symmetry of the hydrogen atom, which makes it possible to find accurate wave functions for the atom in the field of the charged particle responsible for the broadening. The accurate solution is used to analyze the transition from electron-impact broadening to statistical broadening in the wing of the line. Considerable attention is given to the thermal motion of the ions and the related ``reduced mass effect'' that manifests itself at the centers of the Balmer lines. The basic experimental results that make it possible to test the conclusions of existing theory and to discern new problems are presented. The quantum formulation of the line-broadening problem is discussed, as well as line broadening for hydrogenlike ions.