In this talk, we will discuss model reduction techniques that can be used to construct fast approximation algorithms for shape optimization problems. The goal is to obtain approximate solutions at run times that are independent of the resolution of the discrete shapes to be optimized. As applications we will discuss methods for real-time elasticity-based shape interpolation and the processing of curves in shape spaces in which a shape is a single point. Finally, I will outline the concept of compressed vibration modes of elastic bodies, which in contrast to the natural vibration modes, are localized ("sparse") deformations.