Transport-based techniques for signal and data analysis have recently received increased interest. Given their ability to provide accurate generative models for signal intensities and other data distributions, they have been used in a variety of applications, including content-based retrieval, cancer detection, image superresolution, and statistical machine learning, to name a few, and they have been shown to produce state-of-the-art results. Moreover, the geometric characteristics of transport-related metrics have inspired new kinds of algorithms for interpreting the meaning of data distributions. Here, we provide a practical overview of the mathematical underpinnings of mass transport-related methods, including numerical implementation, as well as a review, with demonstrations, of several applications. Software accompanying this article is available.