The paper extends the theory of spectral submanifolds (SSMs) to non-autonomous dynamical systems that are either weakly forced or slowly varying. The authors construct time-dependent SSMs under these assumptions, which are normally hyperbolic and thus persist for larger forcing and faster time dependence. They derive formal asymptotic expansions that approximate SSMs and their aperiodic anchor trajectories under nonresonance conditions, even for stronger, faster, or discontinuous forcing. This provides a mathematically justified model reduction technique for non-autonomous physical systems with moderate time dependence. The existence, persistence, and computation of temporally aperiodic SSMs are illustrated through mechanical examples under chaotic forcing. The paper addresses the limitations of existing SSM theory in handling systems with general time dependence, such as impulsive, chaotic, or discontinuous forcing, and offers a solution by extending SSM theory to temporally aperiodic dynamical systems. The results are applicable to various fields, including structural dynamics, fluid-structure interactions, and control problems.The paper extends the theory of spectral submanifolds (SSMs) to non-autonomous dynamical systems that are either weakly forced or slowly varying. The authors construct time-dependent SSMs under these assumptions, which are normally hyperbolic and thus persist for larger forcing and faster time dependence. They derive formal asymptotic expansions that approximate SSMs and their aperiodic anchor trajectories under nonresonance conditions, even for stronger, faster, or discontinuous forcing. This provides a mathematically justified model reduction technique for non-autonomous physical systems with moderate time dependence. The existence, persistence, and computation of temporally aperiodic SSMs are illustrated through mechanical examples under chaotic forcing. The paper addresses the limitations of existing SSM theory in handling systems with general time dependence, such as impulsive, chaotic, or discontinuous forcing, and offers a solution by extending SSM theory to temporally aperiodic dynamical systems. The results are applicable to various fields, including structural dynamics, fluid-structure interactions, and control problems.