This PhD aims to introduce new techniques to generative modelling from applications of optimal transport. Optimal transport is a rich area of Mathematics which studies how to move mass from one location to another in the most efficient manner possible. This notion was originally formulated to study how to allocate physical resources from supply to demand but in recent history it was been found to be massively applicable to more abstract transport problems. An emerging area within generative modelling is the use of the Wasserstein distance as a distance between probability distributions, where the Wasserstein distance is a core object of study in optimal transport. There has been some progress due to this introduction, however there is still significant amounts of theory that have not been explored in this context yet. The first piece of preliminary work in this proposal examines how to enforce Lipschitz continuity in a Wasserstein GAN and what impact this has on performance. The other component of the work derives a novel regulariser to the Wasserstein GAN architecture to enhance stability and perform gradient updates in Wasserstein space. The future directions for this PhD are to develop the existing work further and also examine the theoretical nature of generative models in an optimal transport context. This will involve Wasserstein metrics, approximation via the Sinkhorn distance and finding new techniques to follow Wasserstein gradient flows.