Are there ‘basic’ tastes and, if so, how many are there? While, to date, this question has mostly been addressed by sensory scientists, it would seem ripe for contemporary philosophical consideration (i.e., after Plato and Aristotle’s early discussion of the matter). Consider only the fact that the majority of those scientists who have written on the subject appear unable to make up their minds as to whether or not umami should be classed as a basic taste, alongside sweet, bitter, sour, and salty. The emergence of various other taste qualities in recent years, such as fatty acid (or ‘oleogustus’) and kokumi, has led (at least one group of) researchers to add another category, namely ‘alimentary’ tastes alongside the so-called basic tastes. Several different lines of evidence have been used to support the existence of basic tastes, including phenomenological (i.e., introspection or perception), cross-cultural linguistic/anthropological, neurophysiological, and neuroimaging data. Perhaps unsurprisingly, though, the answers provided sometimes differ. In this narrative historical review, we critically evaluate the evidence focusing, in particular, on the problematic case of umami, as well as investigating whether an embodied account of basic taste can be supported. Based on the reviewed evidence, we argue for a three-tiered interpretation of ‘basic’, ‘possibly basic’, and ‘ancillary taste qualities’.