We study the use of amortized optimization to predict optimal transport (OT) maps from the input measures, which we call Meta OT. It is useful when repeatedly solving similar OT problems between different measures because it leverages the knowledge and information present from past problems to rapidly predict and solve new problems. Otherwise, standard methods ignore the knowledge of the past solutions and suboptimally re-solve each problem from scratch. We demonstrate that Meta OT models surpass the standard convergence rates of log-Sinkhornsolvers in the discrete setting and convex potentials in the continuous setting. We evaluate on transport settings between images and spherical data, and show significant improvement in the computational time of standard OT solvers.