Reduced-order modeling is among the leading theoretical and computational challenges in mechanics, either of fluids, solids or their interactions. Direct extractions of nonlinear behavior (e.g., bifurcations, forced responses, state transitions, non-trivial attractors, turbulence) in such high dimensional complex, dynamical systems are computationally demanding, if not unattainable. Both in numerics or experiments, data-driven reduction constitutes an effective approach to obtain tractable models that capture nonlinearizable dynamics. Not only are these models are used for the understanding nonlinear phenomena, but they can also be exploited for design optimization and efficient controllability. Here we present an approach that extracts explicit nonlinear models from data exploiting spectral submanifolds (SSMs) theory. These manifolds govern the asymptotic dynamics and lay the foundations of our exact reduced-order modeling approach, which we call Spectral Submanifold Reduction (SSMR). Via decaying trajectory data from experiments or simulation, we identify the relevant low-dimensional attracting SSM in the observables space and characterize its reduced-dynamics and geometry. SSMR reveals to accurately generalize in predicting trajectories outside its training range or even forced regimes. We illustrate the capabilities of SSMR on examples in fluids dynamics, solid dynamics, as well as their interactions. We consider reduced-order models from direct numerical simulation of the Navier Stokes equations (e.g., vortex shedding of the flow past a cylinder, state transitions in plane Couette flow), and from finite element models of vibrations in beams, plates and MEMS devices. Using experimental data, we apply SSMR to measurements of an inverted flag in a water tunnel and of liquid sloshing in a tank, where we predict resonant forced responses when training on decaying (unforced) oscillations.