The optimal transport (OT) problem is often described as that of finding the most efficient way of moving a pile of dirt from one configuration to another. Once stated formally, OT provides extremely useful tools for comparing, interpolating and processing objects such as distributions of mass, probability measures, histograms or densities. This talk is an up-to-date tutorial on a selection of topics in OT. In the first part, I will give an intuitive description of OT, its behavior and basic properties. I will also explain a useful extension of the theory to deal with unnormalized distributions of mass. In the second part, I will introduce state-of-the-art numerical methods for solving OT related problems, namely scaling algorithms based on entropic regularization.