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Dynamic Monte Carlo methods are widely used in scientific and engineering computing.  For certain common tasks, e.g., computing sensitivities with respect to parameter variations, Monte Carlo can unfortunately be rather expensive.  In this talk, I will report on 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 potentially applicable to nonequilibrium steady-state calculations in statistical physics.