Much like the term “computational linguistics”, the term “computational phonology” has come to mean different things to different people. Research grounded in a variety of methodologies and formalisms can be included in its scope. The common thread of the research that falls under this umbrella term is the use of computational methods to investigate questions of interest in phonology, primarily how to delimit the set of possible phonological patterns from the larger set of “logically possible” patterns and how those patterns are learned. Computational phonology arguably began with the foundational result that Sound Pattern of English (SPE) rules are regular relations (provided they can’t recursively apply to their own structural change), which means they can be modeled with finite-state transducers (FSTs) and that a system of ordered rules can be composed into a single FST. The significance of this result can be seen in the prominence of finite-state models both in theoretical phonology research and in more applied areas like natural language processing and human language technology. The shift in the field of phonology from rule-based grammars to constraint-based frameworks like Optimality Theory (OT) initially sparked interest in the question of how to model OT with FSTs and thereby preserve the noted restriction of phonology to the complexity level of regular. But an additional point of interest for computational work on OT stemmed from the ways in which its architecture readily lends itself to the development of learning algorithms and models, including statistical approaches that address recognized challenges such as gradient acceptability, process optionality, and the learning of underlying forms and hidden structure. Another line of research has taken on the question of to what extent phonology is not just regular, but subregular, meaning describable with proper subclasses of the regular languages and relations. The advantages of subregular modeling of phonological phenomena are argued to be stronger typological explanations, in that the computational properties that establish the subclasses as properly subregular restrict the kinds of phenomena that can be described in desirable ways. Also, these same restrictions lead directly to provably correct learning algorithms. Once again this work has made extensive use of the finite-state formalism, but it has also employed logical characterizations that more readily extend from strings to non-linear phenomena such as autosegmental representations and syllable structure.