Generative AI has received much attention in the image and language domains, with the transformer neural network continuing to dominate the state of the art. Application of these models to time series generation is less explored, however, and is of great utility to machine learning, privacy preservation, and explainability research. The present survey identifies this gap at the intersection of the transformer, generative AI, and time series data, and reviews works in this sparsely populated subdomain. The reviewed works show great variety in approach, and have not yet converged on a conclusive answer to the problems the domain poses. GANs, diffusion models, state space models, and autoencoders were all encountered alongside or surrounding the transformers which originally motivated the survey. While too open a domain to offer conclusive insights, the works surveyed are quite suggestive, and several recommendations for best practice, and suggestions of valuable future work, are provided.