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Abstract Structured waves are ubiquitous for all areas of wave physics, both classical and quantum, where the wavefields are inhomogeneous and cannot be approximated by a single plane wave. Even the interference of two plane waves, or of a single inhomogeneous (evanescent) wave, provides a number of nontrivial phenomena and additional functionalities as compared to a single plane wave. Complex wavefields with inhomogeneities in the amplitude, phase, and polarization, including topological structures and singularities, underpin modern nanooptics and photonics, yet they are equally important, e.g., for quantum matter waves, acoustics, water waves, etc. Structured waves are crucial in optical and electron microscopy, wave propagation and scattering, imaging, communications, quantum optics, topological and non-Hermitian wave systems, quantum condensed-matter systems, optomechanics, plasmonics and metamaterials, optical and acoustic manipulation, and so forth. This Roadmap is written collectively by prominent researchers and aims to survey the role of structured waves in various areas of wave physics. Providing background, current research, and anticipating future developments, it will be of interest to a wide cross-disciplinary audience. 

Synthetic photonic materials created by engineering the profile of refractive index or gain/loss distribution, such as negative-index metamaterials or parity-time-symmetric structures, can exhibit electric and magnetic properties that cannot be found in natural materials, allowing for photonic devices with unprecedented functionalities. In this article, we discuss two directions along this line—non-Hermitian photonics and topological photonics—and their applications in nonreciprocal light transport when nonlinearities are introduced. Both types of synthetic structures have been demonstrated in systems involving judicious arrangement of optical elements, such as optical waveguides and resonators. They can exhibit a transition between different phases by adjusting certain parameters, such as the distribution of refractive index, loss, or gain. The unique features of such synthetic structures help realize nonreciprocal optical devices with high contrast, low operation threshold, and broad bandwidth. They provide promising opportunities to realize nonreciprocal structures for wave transport.