This paper discusses problems associated with the expressions for the angular momentum density and the flux density of the angular momentum of an electromagnetic wave within the framework of classical electrodynamics. We show that the formal application of the quantum relationships for a photon to the calculation of the flux density of the angular momentum yields false results. For a plane electromagnetic wave this quantity equals zero despite the opinion established in the literature. This conclusion does not contradict the possibilities of angular momentum transfer in the interaction of a circularly polarized wave with material objects. In particular, in the absorption of such a wave in a plasma, angular momentum is partly transferred to the electrons, and closed quasistationary electric currents arise. The process of generation of the magnetic field excited here possesses a number of distinctions from the inverse Faraday effect known in the literature.