We discuss several geometric approaches to the study of data sets in high-dimensional spaces that are assumed to have low-intrinsci dimension. On the one hand, we discuss diffusion geometry type of approaches, based on constructing proximity graphs between the data points in high dimensions, and using diffusion processes on such graphs to construct coordinates for the data and perform learning tasks. On the other hand, we discuss novel approaches based on multiscale geometric analysis, based on studying the behavior of local covariance matrices of the data at different scales. We apply this latter approach to intrinsic dimension estimation, multiscale dictionary learning, and density estimation.