Deep generative models have seen a meteoric rise in capabilities across a wide array of domains, ranging from natural language and vision to scientific applications such as precipitation forecasting and molecular generation. However, a number of important applications focus on data which is inherently infinite-dimensional, such as time-series, solutions to partial differential equations, and audio signals. This relatively under-explored class of problems poses unique theoretical and practical challenges for generative modeling. In this talk, we will explore recent developments for infinite-dimensional generative models, with a focus on diffusion-based methodologies.