Statistical models with latent structure have a history going back to the 1950s and have seen widespread use in the social sciences and, more recently, in computational biology and in machine learning. Here we study the basic latent class model proposed originally by the sociologist Paul F. Lazarfeld for categorical variables, and we explain its geometric structure. We draw parallels between the statistical and geometric properties of latent class models and we illustrate geometrically the causes of many problems associated with maximum likelihood estimation and related statistical inference. In particular, we focus on issues of non-identifiability and determination of the model dimension, of maximization of the likelihood function and on the effect of symmetric data. We illustrate these phenomena with a variety of synthetic and real-life tables, of different dimension and complexity. Much of the motivation for this work stems from the "100 Swiss Francs" problem, which we introduce and describe in detail.