Stoquastic Hamiltonians are local Hamiltonians with ground states and Gibbs states on the border between quantum and classical computational complexity.   This talk will focus on classically simulating Gibbs states of stoquastic Hamiltonians using Markov chain Monte Carlo methods, including polynomial-time algorithms for 1D stoquastic Hamiltonians (arXiv:1812.02144) at any fixed temperature, and generalized quantum Ising models at high temperature (arXiv:2002.02232).   The talk will also describe open problems and challenges, including simulation of more general stoquastic Hamiltonians at high temperature, as well as simulation of ground state stoquastic adiabatic computation.