We study the problem of optimal transport in tropical geometry and define the Wasserstein-p distances in the continuous metric measure space setting of the tropical projective torus. We specify the tropical metric—a combinatorial metric that has been used to study of the tropical geometric space of phylogenetic trees—as the ground metric and study the cases of p=1,2 in detail. The case of p=1 gives an efficient computation of the infinitely-many geodesics on the tropical projective torus, while the case of p=2 gives a form for Fréchet means and a general inner product structure. Our results also provide theoretical foundations for geometric insight a statistical framework in a tropical geometric setting. We construct explicit algorithms for the computation of the tropical Wasserstein-1 and 2 distances and prove their convergence. Our results provide the first study of the Wasserstein distances and optimal transport in tropical geometry. Several numerical examples are provided.