Solving Maxwell's equations in cylindrical coordinates yields states in quantum theory with definite values of energy hω, longitudinal momentum hkz, and total angular momentum projection hm on the z-axis (where h is the Planck constant). Experimentally, values of up to hm ∼ 104 h have been obtained for the last quantity. The wave front of such states is like a meat grinder screw, with the lines of force of the Poynting vector representing the screw line. Such states differ from plane waves by the nonzero orbital angular momentum projection on the direction of motion, and from spherical waves, by the definite direction of motion. For brevity, these states are referred to as 'twisted photons'. In this paper, recent experimental and theoretical results on twisted photons are reviewed, to which the present authors actively contributed. Detailed discussion is given to recent experiments on the production of high-intensity beams of terahertz (wavelength: 140 μm) twisted photons performed on the Novosibirsk free electron laser at the Budker Institute of Nuclear Physics, SB RAS. Recent theoretical work on the interaction of twisted photons with atoms is summarized. Due to their extra degree of freedom-the projection of the total angular momentum on the direction of motion-twisted photons represent a novel research tool of potentially wide application in physics.