We review the ideas, algorithms, and numerical performance of manifold-based machine learning and dimension reduction methods. The representative methods include locally linear embedding (LLE), ISOMAP, Laplacian eigenmaps, Hessian eigenmaps, local tangent space alignment (LTSA), and charting. We describe the insights from these developments, as well as new opportunities for both researchers and practitioners. Potential applications in image and sensor data are illustrated. This chapter is based on an invited survey presentation that was delivered by Huo at the 2004 INFORMS Annual Meeting, which was held in Denver, CO, USA.