The theory of optimal transport (OT) gives rise to distance measures between probability distributions that take the geometry of the underlying space into account. OT is often used in the analysis of point cloud data, for example in domain adaptation problems, computer graphics, and trajectory analysis of single-cell RNA-Seq data. However, from a statistical perspective, straight-forward plug-in estimators for OT distances and couplings suffer from the curse of dimensionality in high dimensions. One way of alleviating this problem is to employ regularized statistical procedures, either by changing the transport objective or exploiting additional structure in the underlying probability distributions or ground truth couplings. In this talk, I will outline the problem and give an overview of recent solution approaches, in particular those employing entropically regularized optimal transport or imposing smoothness assumptions on the ground truth transport map.