Inspired by the success of compressive sensing, the last three years have seen an explosion of research in the theory of low-rank modeling. By now, we have results stating that it is possible to recover certain low-rank matrices from a minimal number of entries -- or of linear functionals -- by tractable convex optimization. We further know that these methods are robust vis a vis additive noise and even outliers. In a different direction, researchers have developed computationally tractable methods for clustering high-dimensional data points that are assumed to be drawn from multiple low-dimensional linear subspaces. This talk will survey some exciting results in these areas.