Invariant manifolds provide a rigorous foundation for model reduction and the analysis of complex dynamical behavior in nonlinear systems. Recent advances have enabled direct and nonintrusive computation of invariant manifolds and their reduced dynamics for high-dimensional finite-element models. In this talk, we discuss how these developments facilitate efficient nonlinear analysis, design and optimization of mechanical systems dependent on parameters. We demonstrate the effectiveness of the methodology on a range of high-fidelity numerical models, where the resulting reduced-order models enable accurate analysis of local bifurcations, including buckling, flutter, period-doubling, and internal and parametric resonances, at computational costs far below those of the full system.