Monte Carlo sampling is a powerful approach to computing observables in quantum field theories regularized on a discrete spacetime lattice (LQFT), which is necessary for example to study the non-perturbative behavior of QCD in the low-energy regime. The cost of drawing independent samples is a major bottleneck in such studies. I discuss recent work demonstrating that generative models from the machine learning community can be used to perform Monte Carlo sampling and produce unbiased estimates of observables in LQFT. Our work lays out a framework for exactly encoding translational and gauge symmetries into these models, making training practically viable. I discuss proof-of-principle tests of the method for scalar field theory as well as for U(1) and SU(N) gauge theories in two spacetime dimensions.