In nature, there are many dynamical systems that are reasonably well-understood from the viewpoint of physics, but are computationally slow to simulate with detailed models. The observed transients in many simulations are, however, often not very rich, but exhibit primarily simple, low-dimensional dynamics. Such observations form the cornerstone of model reduction, an interdisciplinary technique used in electrical and aerospace engineering, as well as solid and fluid mechanics, and computer graphics. In my career, I have studied how model reduction can be applied to elasticity problems in computer graphics and animation. I will present my work on model reduction applied to fast simulation, optimal control, optimization, collision detection, sound simulation and interactive design of geometrically and materially complex models undergoing large deformations.