Recently, the adapted Wasserstein (AW) distance has emerged as an attractive alternative to the usual distances between laws of stochastic processes due to its application to dynamic optimisation problems, such as optimal stopping, for example. However, similar to classical Wasserstein distance, the exact value of the AW distance is difficult to determine in general. In this talk, we discuss two methods — a transfer principle, and a discretisation approach — to compute, either explicitly or numerically, the AW distance between the laws of certain stochastic processes, such as mean-square continuous Gaussian processes and stochastic differential equations.