Optimal Transport (OT) problems are a cornerstone of many applications, but solving them is computationally expensive. To address this problem, we propose UNOT (Universal Neural Optimal Transport), a novel framework capable of accurately predicting (entropic) OT distances and plans between discrete measures of variable resolution for a given cost function. UNOT builds on Fourier Neural Operators, a universal class of neural networks that map between function spaces and that are discretization-invariant, which enables our network to process measures of varying sizes. The network is trained adversarially using a second, generating network and a self-supervised bootstrapping loss. We theoretically justify the use of FNOs, prove that our generator is universal, and that minimizing the bootstrapping loss provably minimizes the ground truth loss. Through extensive experiments, we show that our network not only accurately predicts optimal transport distances and plans across a wide range of datasets, but also captures the geometry of the Wasserstein space correctly. Furthermore, we show that our network can be used as a state-of-the-art initialization for the Sinkhorn algorithm, significantly outperforming existing approaches.