This paper extends the theory of spectral submanifolds (SSMs) to general non-autonomous dynamical systems with weak or slowly varying forcing. SSMs are invariant manifolds that describe the internal dynamics of a system on lower-dimensional, attracting invariant sets in phase space. They are particularly useful for non-autonomous systems where the time dependence is moderate in magnitude or speed. The paper addresses the challenge of applying SSM reduction to systems with more general time dependence, such as impulsive, chaotic, or discontinuous forcing, which have been previously inapplicable to SSM methods. The authors derive formal asymptotic expansions for SSMs and their aperiodic anchor trajectories under explicitly verifiable nonresonance conditions. These expansions allow for accurate approximation of SSMs and their trajectories even for stronger, faster, or temporally discontinuous forcing. The results are validated through mechanical examples under chaotic forcing, demonstrating the effectiveness of the new asymptotic formulas for reduced-order modeling. The paper presents two main settings: weak non-autonomous forcing and slowly varying (adiabatic) forcing. For weak forcing, the authors derive recursive formulas for the anchor trajectory and SSMs, showing that these can be extended to larger and faster forcing. For adiabatic forcing, the authors develop a computational algorithm for SSMs that can handle slowly varying systems. The paper also discusses the existence and computation of non-autonomous SSMs under various conditions, including nonresonance and hyperbolicity assumptions. The results show that non-autonomous SSMs can persist under a wide range of forcing conditions, including discontinuous and chaotic forcing. The authors provide formal asymptotic expansions for these SSMs, which can be used to compute reduced-order models for non-autonomous systems. The paper concludes with a discussion of the implications of these results for the broader field of dynamical systems, emphasizing the importance of SSM reduction for systems with general aperiodic time dependence. The results are validated through numerical examples and are shown to be effective for both equation-driven and data-driven applications.This paper extends the theory of spectral submanifolds (SSMs) to general non-autonomous dynamical systems with weak or slowly varying forcing. SSMs are invariant manifolds that describe the internal dynamics of a system on lower-dimensional, attracting invariant sets in phase space. They are particularly useful for non-autonomous systems where the time dependence is moderate in magnitude or speed. The paper addresses the challenge of applying SSM reduction to systems with more general time dependence, such as impulsive, chaotic, or discontinuous forcing, which have been previously inapplicable to SSM methods. The authors derive formal asymptotic expansions for SSMs and their aperiodic anchor trajectories under explicitly verifiable nonresonance conditions. These expansions allow for accurate approximation of SSMs and their trajectories even for stronger, faster, or temporally discontinuous forcing. The results are validated through mechanical examples under chaotic forcing, demonstrating the effectiveness of the new asymptotic formulas for reduced-order modeling. The paper presents two main settings: weak non-autonomous forcing and slowly varying (adiabatic) forcing. For weak forcing, the authors derive recursive formulas for the anchor trajectory and SSMs, showing that these can be extended to larger and faster forcing. For adiabatic forcing, the authors develop a computational algorithm for SSMs that can handle slowly varying systems. The paper also discusses the existence and computation of non-autonomous SSMs under various conditions, including nonresonance and hyperbolicity assumptions. The results show that non-autonomous SSMs can persist under a wide range of forcing conditions, including discontinuous and chaotic forcing. The authors provide formal asymptotic expansions for these SSMs, which can be used to compute reduced-order models for non-autonomous systems. The paper concludes with a discussion of the implications of these results for the broader field of dynamical systems, emphasizing the importance of SSM reduction for systems with general aperiodic time dependence. The results are validated through numerical examples and are shown to be effective for both equation-driven and data-driven applications.