Problems in modelling categorical perception (CP) and attempts to apply signal detection theory (SDT) to CP are reviewed. An approach based on SDT supplemented by a theory of criterion setting is presented. Criterion setting theory (CST) postulates mechanisms that reset the response criterion on each trial, and it accounts for sequential dependencies. A criterion setting model for discrimination is shown to fit data from the literature. The hypothesis that "sharp" category boundaries may arise from the suppression of noise caused by intertrial dependencies was examined in an experiment on the identification of [ba] and [pa] syllables, and tone combinations of varying tone-onset time. However, it was shown that both positive and negative intertrial dependencies were present. They could be fitted by the criterion-setting model; in this respect, CP resembles standard psychophysical judgements. Examination of the psychometric functions from the two CP tasks shows that they are not normal ogives, as in standard psychophysical tasks: these curves are steeper centrally and flatter at the extremes than a Gaussian ogive; we describe them as "hypersigmoid". The description of CP identification functions as hypersigmoid provides a new, qualitative characterization of the "sharp" category boundaries traditionally claimed for CP. Their causation remains to be determined.