Broadly speaking, computational phonology encompasses a variety of techniques and goals (see Daland 2014 for a survey). In this introduction we would like to highlight three aspects of current work in computational phonology: data science and model comparison, modelling phonological phenomena using computational simulations, and characterising the computational nature of phonological patterning with theorems and proofs. Papers in this thematic issue illustrate all three of these trends, and sometimes more than one of them. The way we group them in this introduction is meant to highlight the similarities between them, and not to diminish the importance of their other contributions. As we discuss these areas, we also highlight important conceptual issues that we believe are often overlooked.