  We extend the theory of spectral submanifolds (SSMs) to general
non-autonomous dynamical systems that are either weakly forced or slowly
varying. Examples of such systems arise in structural dynamics, fluid-structure
interactions and control problems. The time-dependent SSMs we construct under
these assumptions are normally hyperbolic and hence will persist for larger
forcing and faster time dependence that are beyond the reach of our precise
existence theory. For this reason, we also derive formal asymptotic expansions
that, under explicitly verifiable nonresonance conditions, approximate SSMs and
their aperiodic anchor trajectories accurately for stronger, faster or even
temporally discontinuous forcing. Reducing the dynamical system to these
persisting SSMs provides a mathematically justified model reduction technique
for non-autonomous physical systems whose time dependance is moderate either in
magnitude or speed. We illustrate the existence, persistence and computation of
temporally aperiodic SSMs in mechanical examples under chaotic forcing.
