The talk will describe how randomized algorithms can effectively, accurately, and reliably solve linear algebraic problems that are omnipresent in scientific computing and in data analysis. We will focus on techniques for low rank approximation, since these methods are particularly simple and powerful, and are well understood mathematically. The talk will also briefly survey how randomized techniques can be applied to construct compressed representations of global operators such as solution operators to elliptic PDEs, boundary-to-boundary operators such as the Dirichlet-to-Neumann map, and time evolution operators of parabolic PDEs.