Physics modeling of complex dynamical nonlinear systems is challenging, and the analysis and design of dynamics of such complex systems face the curse of high-dimensionality. Therefore, it is important to develop low-dimensional, reliable reduced-order models (ROMs) for them. Spectral submanifolds (SSMs) have emerged as a powerful tool for constructing ROMs for complex systems. SSMs are low-dimensional attracting invariant manifolds, and the associated SSM-based ROMs are low-dimensional yet exact. They can be obtained in equation-driven and data-driven settings and have been successfully applied to nonlinear vibrations, fluid dynamics, and control of soft robots. We present a review of recent advances in the theory of SSMs, model reduction techniques via SSMs, and the various applications of SSM-based reductions. We conclude this article with an outlook on future developments.