Abstract: In this talk we will explore the connection between the two seemingly unrelated concepts appearing in the title. Phase transitions occur when a system undergoes an abrupt change in behaviour as a consequence of a small change in parameters. While phase transitions are evidently observed in the physical world (e.g., water freezing or evaporating), they are also ubiquitous in mathematical problems studied in statistical mechanics, probability, combinatorics and computer science. Finitary coding is an ergodic-theoretic notion which is concerned with the expressibility of certain processes in terms of other processes. After introducing these two concepts, we will discuss some results which establish close links between them.