We discuss the properties of twisted waves in a plasma, carrying orbital angular momentum. New wave phenomena associated with these waves can be explored in plasma physics. Angular momentum states of electrostatic and electromagnetic fields are introduced. These states can be excited by various methods, including plasma rotation and nonlinear wave mixing. The exchange of angular momentum between electromagnetic and electrostatic waves in a plasma can be achieved by stimulated Raman and Brillouin scattering processes. The description of such processes is based on nonlinearly coupled paraxial equations. Other new effects due to wave vorticity can be observed in plasmas. In particular, intense magnetic fields associated with the inverse Faraday effect can be generated by linearly polarized laser beams, thus changing our traditional view of this effect.