Structured Latent Attribute Models (SLAMs) are popular statistical tools for developing diagnostic-based assessments in education, psychology, and other social and behavioral sciences. SLAMs can be viewed as a family of restricted discrete latent variable models, which assume that multiple discrete latent attributes explain the dependence of observed variables in a highly structured fashion. Though widely used, such structured latent class models often suffer from nonidentifiability due to the models' discrete nature and complex restricted structure. The first part of this talk introduces our recent identifiability results on SLAMs by considering both strict and partial identifiability of the model parameters. The developed identifiability conditions only depend on the design matrix and are easily checkable, which provides useful practical guidelines for designing statistically valid diagnostic tests. The second part of the talk further discusses likelihood-based approaches to estimate the latent structures and the model parameters.