Optical vortices, characterized by their central singularities and spiral phase wavefront, have gained widespread attention in light manipulation and diverse applications. With their topological order being extended from integer to fraction, more unique properties have sprung out, such as continuous topological charges, fractional spiral-phase, expanded orbital angular momentum spectrum, and slit openings. These features have facilitated applications in areas such as optical tweezers, direction-selective edge enhancement imaging, robust rotational Doppler metrology, free-space communication, displacement sensing, and high-dimensional quantum entanglement. In recent years, advances in metasurface-based optical control, spatiotemporal pulse shaping, holographic imaging, and deep learning have spurred rapid innovation in the modeling, generation, and measurement of fractional vortex beams. This review begins with the fundamental theory of fractional vortex beams and surveys the latest developments in this rapidly evolving field. The scope of research has also expanded beyond optical vortices to include acoustic vortices. The growing interest in fractional vortex beams, coupled with ongoing technological innovations, is expected to pave the way for further advancements in this promising area of research.

Topology, a cornerstone of modern condensed matter physics, has in the past decade played a crucial role in diverse wave systems. As a powerful wave system, light can be sculpted into an even richer variety of topological structures, including vortices, skyrmions, Möbius strips, etc., leading to advanced photonic technologies from optical trapping to imaging, across quantum and classical regimes. A recent breakthrough demonstrated that topologically structured water waves can manipulate particles with intricate spin-orbital motion, and similar principles have enabled topological control in acoustofluidics, opening new insights in wave-matter interactions. Therefore, we argue that topological light waves possess an analogous potential, offering a route beyond the scalar-field limitations of conventional optical tweezers and establishing a new paradigm of multidimensional, vectorial control over matter. This article starts with brief introductions to optical tweezer technologies and topological light waves, then focuses on their emerging combination: particle trapping and sorting. It follows with perspectives on how topologies can couple to new degrees of freedom for manipulating complex particle motions previously inaccessible, and finally discusses potential applications.