Lagrangian and Hamiltonian mechanics are foundational modeling approaches in diverse areas such as structural mechanics, aerospace engineering, wave propagation, biomedical engineering, high-energy physics, quantum mechanics, solid-state physics, and soft robotics. These systems exhibit physically interpretable quantities such as momentum, energy, or vorticity; the behavior of these quantities in numerical simulation provides an important measure of accuracy of the model. For the data-driven modeling and simulation such high-dimensional and structured complex dynamical systems, it is essential to first reduce the dimensionality to manageable (reduced) dimensions and then to incorporate the Lagrangian and Hamiltonian structure into the learning framework. This talk gives an overview of a few recently developed approaches for learning structure-preserving reduced-order models, with applications to structural mechanics, soft robotics, and wave propagation. With much fewer training data than non-structured learning methods require, the Lagrangian and Hamiltonian learned reduced models provides stability and robustness, and have long-term predictive accuracy.