In applications of optimal transport in statistics, it is paramount to be able to obtain good upper and lower bounds on the Wasserstein distance between probability measures. This chapter describes tools to bound the Wasserstein distance. To do so, we heavily employ the primal and dual formulations of optimal transport. As a primary application, we consider a quantitative form of the Wasserstein law of large numbers, which is the statement that if \(\mu _n\) is an empirical measure consisting of n i.i.d. samples from a probability measure \(\mu \) , then \(E W_p(\mu _n, \mu ) \to 0\) as \(n \to \infty \) .