We consider the problem of representation learning of probability distributions in Wasserstein space. We introduce a general barycentric coding model in which data are represented as Wasserstein-2 ($W_{2}$) barycenters of a set of fixed reference measures. Leveraging the Riemannian structure of $W_{2}$-space, we develop a tractable optimization program to learn the barycentric coordinates when given access to the densities of the underlying measures. We provide a consistent statistical procedure for learning these coordinates when the measures are accessed only by i.i.d. samples. Our consistency results and algorithms exploit entropic regularization of optimal transport maps, thereby allowing our barycentric modeling approach to scale efficiently. Throughout the talk, applications to image and natural language processing demonstrate the efficacy of our geometric methods.