Multinomial processing tree (MPT) models such as the single high-threshold, double high-threshold, and low-threshold models are discrete-state decision models that map internal cognitive events onto overt responses. The apparent benefit of these models is that they provide independent measures of accuracy and response bias, a claim that has motivated their frequent application in many areas of psychological science including perception, item and source memory, social cognition, reasoning, educational testing, eyewitness testimony, and psychopathology. Before appropriate conclusions about a given analysis can be drawn, however, one must first confirm that the model's assumptions about the underlying structure of the data are valid. The current review outlines the assumptions of several popular MPT models and assesses their validity using multiple sources of evidence, including receiver operating characteristics, direct model fits, and experimental tests of qualitative predictions. We argue that the majority of the evidence is inconsistent with these models and that, instead, the evidence supports continuous models such as those based on signal detection theory (SDT). Hybrid models that incorporate both SDT and MPT processes are also explored, and we conclude that these models retain the limitations associated with their threshold model predecessors. The potentially severe consequences associated with using an invalid model to interpret data are discussed, and a simple tutorial and model-fitting tool is provided to allow implementation of the empirically supported SDT model. (PsycINFO Database Record (c) 2013 APA, all rights reserved).