Sampling is a fundamental and widespread algorithmic primitive that lies at the heart of Bayesian inference and scientific computing, among other disciplines. Recent years have seen a flood of works aimed at laying down the theoretical underpinnings of sampling, in analogy to the fruitful and widely used theory of convex optimization. In this talk, I will discuss some of my work in this area, focusing on new convergence guarantees obtained via a proximal algorithm for sampling, as well as a new framework for studying the complexity of non-log-concave sampling.