Optimal transportation distances are a fundamental family of parameterized distances for histograms in the probability simplex. Despite their appealing theoretical properties, excellent performance and intuitive formulation, their computation involves the resolution of a linear program whose cost is prohibitive whenever the histograms' dimension exceeds a few hundreds. We propose in this work a new family of optimal transportation distances that look at transportation problems from a maximum-entropy perspective. We smooth the classical optimal transportation problem with an entropic regularization term, and show that the resulting optimum is also a distance which can be computed through Sinkhorn's matrix scaling algorithm at a speed that is several orders of magnitude faster than that of transportation solvers. We also report improved performance on the MNIST benchmark problem over competing distances.