Machine learning has had a major impact on data compression over the last decade and opened up many new theoretical and applied fields of inquiry. This paper describes one such direction -- relative entropy coding -- which focuses on constructing stochastic codes, mainly as an alternative to quantisation and entropy coding in lossy source coding. Our primary aim is to provide a broad overview of the topic, with an emphasis on the computational and practical aspects currently missing from the literature. Our goal is threefold: for the curious reader, we aim to provide an intuitive picture of the field and convince them that relative entropy coding is a simple yet exciting emerging field in data compression research. For a reader interested in applied research on lossy data compression, we provide an account of the most salient contemporary applications. Finally, for the reader who has heard of relative entropy coding but has never been quite sure what it is or how the algorithms fit together, we hope to illustrate how simple and elegant the underlying constructions are.