Dynamic Monte Carlo methods are widely used in scientific and engineering computing. In this talk, I will report on some recent efforts to accelerate dynamic Monte Carlo calculations using Markov couplings. Specifically, I will describe coupling-based algorithms for computing sensitivities of stationary averages for stochastic differential equations, and discuss some of the strengths and limitations of these algorithms. Time permitting, I will also discuss how these ideas may be applied to nonequilibrium steady-state calculations in statistical physics.