Rapid progress in miniaturizing vortex devices is driven by their integration with optical sensing, micromanipulation, and optical communications in both classical and quantum realms. Many such efforts are usually associated with on-chip micro- or nanoscale structures in real space and possess a static orbital angular momentum. Recently, a new branch of singular optics has emerged that seeks phase singularities in multiple dimensions, realizing vortex beams with compact nanodevices. Here, we review the topological phase singularities in real space, momentum space, and the spatiotemporal domain for generating vortex beams; discuss recent developments in theoretical and experimental research for generation, detection, and transmission of vortex beams; and provide an outlook for future opportunities in this area, ranging from fundamental research to practical applications.