Complex data analysis is a central topic of modern statistics and learning systems which is becoming of broader interest with the increasing prevalence of high-dimensional data. The challenge is to develop statistical models and autonomous algorithms that are able to discern knowledge from raw data, which can be achieved through clustering techniques, or to make predictions of future data via classification techniques. Latent data models, including mixture model-based approaches, are among the most popular and successful approaches in both supervised and unsupervised learning. Although being traditional tools in multivariate analysis, they are growing in popularity when considered in the framework of functional data analysis (FDA). FDA is the data analysis paradigm in which each datum is a function, rather than a real vector. In many areas of application, including signal and image processing, functional imaging, bioinformatics, etc., the analyzed data are indeed often available in the form of discretized values of functions, curves, or surfaces. This functional aspect of the data adds additional difficulties when compared to classical multivariate data analysis. We review and present approaches for model-based clustering and classification of functional data. We present well-grounded statistical models along with efficient algorithmic tools to address problems regarding the clustering and the classification of these functional data, including their heterogeneity, missing information, and dynamical hidden structures. The presented models and algorithms are illustrated via real-world functional data analysis problems from several areas of application.