Structured light, especially beams carrying orbital angular momentum (OAM), has gained much interest due to its unique amplitude and phase structures. In terms of communication systems, multiple orthogonal OAM beams can be potentially utilized for increasing link capacity in different scenarios. This review describes challenges, advances, and perspectives on different aspects of the OAM-based optical communications, including (a) OAM generation/detection and (de)multiplexing, (b) classical free-space optical communication links, (c) fiber-based communication links, (d) quantum communication links, (e) OAM-based communications in different frequency ranges, (f) OAM-based communications using integrated devices, and (g) novel structured beams for communications.