I will discuss a family of recently developed stochastic techniques for linear algebra problems involving very large matrices.  These methods can be used to, for example, solve linear systems, estimate eigenvalues/vectors, and apply a matrix exponential to a vector, even in cases where the desired solution vector is too large to store.  The first incarnations of this idea appear for dominant eigenproblems arising in statistical physics and in quantum chemistry and were inspired by the real space diffusion Monte Carlo algorithm which has been used to compute chemical ground states since the 1970's.  I will discuss our own general framework for fast randomized iterative linear algebra as well share a very partial explanation for their effectiveness.  I will also report on the progress of an ongoing collaboration aimed at developing fast randomized iterative schemes specifically for applications in quantum chemistry.  This talk is based on joint work with Lek-Heng Lim, Timothy Berkelbach, Sam Greene, and Rob Webber.