In recent years a variety of spectral and geometry-based methods have become popular for various tasks of machine learning, such as dimensionality reduction, clustering and semi-supervised learning. These methods use a model of data as a probability distribution on a manifold, or, more generally a mixture of manifolds. In the talk I will discuss some of these methods and recent theoretical results on their convergence. I will also talk about how spectral methods can be used to learn mixtures of Gaussian distributions, which may be considered the simplest case of multi-manifold data modeling.