This thesis develops novel Bayesian methodologies for statistical modelling of heavy-tailed data. Heavy tails are often found in practice, and yet they are an Achilles heel of a variety of main-stream random probability measures such as the Dirichlet process. The first contribution of this thesis is the study of random probability measures that can be used to model heavy-tailed data, focusing on the characterization of the tails of the so-called Pitman–Yor process, which includes the Dirichlet process as a particular case. We show that the right tail of a Pitman–Yor process, known as the stable law process, is heavy-tailed, provided that the centering distribution is itself heavy-tailed. Empowered by this finding we developed two classes of heavy-tailed mixture models and assessed their relative merits. Multivariate extensions of the proposed heavy-tailed mixtures are also devised along with a predictor-dependent version so to learn about the effect of covariates on a multivariate heavy-tailed response. Another contribution of the thesis is a framework for learning about the frequency and magnitude of extreme values in the joint tail as well as in other risk sets induced by a suitable aggregation of variables on a common scale. The proposed framework reveals explicit links between nonstationary multivariate extremes and heteroscedastic extremes, and hence can be used to track the dynamics governing the degree of association between the extremes of a random vector over time. Bayesian inference methods are conducted for the two targets of interest: the structure scedasis function and the coefficient of tail dependence for structure processes. To learn about the structure scedasis function from data, we resort to a Bayesian nonparametric approach that defines a prior in the space of structure scedasis functions. A battery of numerical experiments is used to study the finite sample performance of all methodological developments made in this thesis. Data from the fields of neuroscience as well as from finance are used to showcase the proposed methodologies in practice.