This review examines the numerous studies performed in recent years on atomic processes involving ``cold'' hydrogen atoms in the gas phase and at a surface. The term ``cold'' atom denotes the range of relative collision energies $E$ < 10 meV (correspondingly, ``ultracold'' atoms have $E\lesssim$ 1 meV). This corresponds to the temperature range of tens (or units) of kelvins. Three main fields of study in which extensive experimental programs have been realized are treated: the study of collision processes involving hydrogen atoms, which are of astrophysical interest; the study of elastic atom-molecule scattering at ultralow energies; and studies on the problem of condensed hydrogen. The fundamental technical means of realization and the results obtained from the studies are discussed.