I will first consider the Busemann function, which naturally defines projections onto geodesic rays in Riemannian manifolds and generalizes the notion of hyperplanes. As the Wasserstein space admits a rich formal Riemannian structure induced by optimal transport metrics, I will then discuss the existence and computation of the Busemann function in this setting. In particular, I will present closed-form expressions in two important cases: one-dimensional probability measures and Gaussian distributions. I will then consider another notion of projection, namely the metric projection, and demonstrate how these projections can be used for transfer learning and geodesic PCA.