  We establish general upper bounds on the Kolmogorov distance between two
probability distributions in terms of the distance between these distributions
as measured with respect to the Wasserstein or smooth Wasserstein metrics.
These bounds generalise existing results from the literature. To illustrate the
broad applicability of our general bounds, we apply them to extract Kolmogorov
distance bounds from multivariate normal, beta and variance-gamma
approximations that have been established in the Stein's method literature.
