Optimal transport has a long history in mathematics which was proposed by Gaspard Monge in the eighteenth century (Monge, 1781). However, until recently, advances in optimal transport theory pave the way for its use in the AI community, particularly for formulating deep generative models. In this paper, we provide a comprehensive overview of the literature in the field of deep generative models using optimal transport theory with an aim of providing a systematic review as well as outstanding problems and more importantly, open research opportunities to use the tools from the established optimal transport theory in the deep generative model domain.