There have been many models and methods proposed for the description and analysis of correlated non-normal data. Our general approach is via the use of HGLMs. Following Lee and Nelder (2001b), we further extend HGLMs in this Chapter to cover a broad class of models for correlated data, and show that many previously developed models appear as instances of HGLMs. Rich classes of correlated patterns in non-Gaussian models can be produced without requiring explicit multivariate generalizations of non-Gaussian distributions. We extend HGLMs by adding an additional feature to allow correlations among the random eﬀects.