Latent structure models involve real, potentially observable variables and latent, unobservable variables. Depending on the nature of these variables, whether they be discrete or continuous, the framework includes various particular types of model, such as factor analysis, latent class analysis, latent trait analysis, latent profile models, mixtures of factor analysers, state-space models and others. The simplest scenario, of a single discrete latent variable, includes finite mixture models, hidden Markov chain models and hidden Markov random field models. The talk will give an overview of the application of maximum likelihood and Bayesian approaches to the estimation of parameters within these models, emphasising especially the fact that computational complexity varies greatly among the different scenarios. In the case of a single discrete latent variable, the issue of assessing its cardinality will be discussed, in the context of questions such as the appropriate number of mixture components to be included in a mixture model, or, in the interests of parsimony, the minimum plausible cardinality of such a latent variable. Techniques such as the EM algorithm, Markov chain Monte Carlo methods and variational approximations will be featured in the talk.