Within the last 40 years the field of phonology has undergone several important theoretical shifts with regard to phonological representation that have now become part of standard practice in the field. The two shifts that will be taken up in this volume have some properties in common—i) both have had the effect of taking phonological analysis “off the page”, that made it necessary to think of phonology in multi-dimensional space, and ii) John Goldsmith has been a major force in bringing about these changes. The first section of this volume has to do with a radical elaboration of the abstract domains (or units of analysis) that come under the purview of phonology, along with a more multi-dimensional approach to considering their role in the system. Autosegmental phonology (Goldsmith, 1976) and feature geometry (Clements, 1981; Sagey, 1986) demonstrated this multidimensionality of phonological representation. The second radical shift that occurred in the mid-1990s has to do with machine learning and the computational techniques that phonologists had begun using to analyze large amounts of data. The empiricist view to linguistics (Goldsmith 2015) makes us rethink what doing linguistics really means. With the ability to employ computational tools that allow for the analysis of larger and larger data sets, the field has shifted from being satisfied to look for key examples that demonstrate a particular generalization to striving for statistical generalizations across large corpora of relevant data.