One of the key challenges in materials physics and chemistry is the accurate computations of systems with stong electron interaction effects. Such effects fundamentally alter the standard independent-electron framework such as Kohn-Sham density functional theory (KS-DFT), and perturbative extensions. To accurately capture the quantum many-body effects, the size of the Hilbert space involved grows exponentially. In this talk we give an introduction to the issues facing this challenge. We then describe recent progress in combining field-theory and Monte Carlo simulations to tackle the problem. The framework can be viewed as a superposition of KS-DFT systems evolving in fluctuating auxiliary fields, which are treated by stochastic sampling. We illustrate the promise of the approach with a few examples in condensed matter physics and quantum chemistry, and discuss opportunities for its improvement and optimization from a mathematical and computational perspective.