Dance choreography for a piece of music is a challenging task, having to be creative in presenting distinctive stylistic dance elements while taking into account the musical theme and rhythm. It has been tackled by different approaches such as similarity retrieval, sequence-to-sequence modeling and generative adversarial networks, but their generated dance sequences are often short of motion realism, diversity and music consistency. In this paper, we propose a Music-to-Dance with Optimal Transport Network (MDOT-Net) for learning to generate 3D dance choreographies from music. We introduce an optimal transport distance for evaluating the authenticity of the generated dance distribution and a Gromov-Wasserstein distance to measure the correspondence between the dance distribution and the input music. This gives a well defined and non-divergent training objective that mitigates the limitation of standard GAN training which is frequently plagued with instability and divergent generator loss issues. Extensive experiments demonstrate that our MDOT-Net can synthesize realistic and diverse dances which achieve an organic unity with the input music, reflecting the shared intentionality and matching the rhythmic articulation. Sample results are found at https://www.youtube.com/watch?v=dErfBkrlUO8.