Low-dimensional modeling with linear subspaces has been the basis for a wide variety of factorization and approximation methods that are useful across science and engineering. In this talk we will view the problem of finding the best-fit linear subspace and the algorithms that solve it through the general lens of optimization. I will first discuss at a high level some of the modifications of this problem explored in signal processing and machine learning for learning low-dimensional structure from data. Then I will share some of my group's recent work on bringing optimization to bear on learning low-dimensional subspaces for reduced order modeling and control of dynamical systems.