Speech perception (SP) most commonly refers to the perceptual mapping from the highly variable acoustic speech signal to a linguistic representation, whether it be phonemes, diphones, syllables, or words. This is an example of categorization, in that potentially discriminable speech sounds are assigned to functionally equivalent classes. In this tutorial, we present some of the main challenges to our understanding of the categorization of speech sounds and the conceptualization of SP that has resulted from these challenges. We focus here on issues and experiments that define open research questions relevant to phoneme categorization, arguing that SP is best understood as perceptual categorization, a position that places SP in direct contact with research from other areas of perception and cognition.