Stein’s method for measuring convergence to a continuous targetdistribution relies on an operator characterizing the target andSteinfactorbounds on the solutions of an associated differential equation.While such operators and bounds are readily available for a diversityof univariate targets, few multivariate targets have been analyzed. Weintroduce a new class of characterizing operators based on Itˆo diffu-sions and develop explicit multivariate Stein factor bounds for anytarget with a fast-coupling Itˆo diffusion. As example applications, wedevelop computable and convergence-determiningdiffusion Stein dis-crepanciesfor log-concave, heavy-tailed, and multimodal targets anduse these quality measures to select the hyperparameters of biasedMarkov chain Monte Carlo (MCMC) samplers, compare random anddeterministic quadrature rules, and quantify bias-variance tradeoffsin approximate MCMC. Our results establish a near-linear relation-ship between diffusion Stein discrepancies and Wasserstein distances,improving upon past work even for strongly log-concave targets. Theexposed relationship between Stein factors and Markov process cou-pling may be of independent interest.