"Diagnostic classification models" (DCM) are frequently promoted by psychometricians as important modelling alternatives for analyzing response data in situations where multivariate classifications of respondents are made on the basis of multiple postulated latent skills. In this review paper, a definitional boundary of the space of DCM is developed, core DCM within this space are reviewed, and their defining features are compared and contrasted with those of other latent variable models. The models to which DCM are compared include unrestricted latent class models, multidimensional factor analysis models, and multidimensional item response theory models. Attention is paid to both statistical considerations of model structure, as well as substantive considerations of model use. (Contains 5 tables and 2 figures.)