This paper unifies and extends results on a class of multivariate Extreme Value (EV) models studied by Hougaard, Crowder, and Tawn. In these models both unconditional and conditional distributions are EV, and all lower-dimensional marginals and maxima belong to the class. This leads to substantial economies of understanding, analysis and prediction. One interpretation of the models is as size mixtures of EV distributions, where the mixing is by positive stable distributions. A second interpretation is as exponential-stable location mixtures (for Gumbel) or as power-stable scale mixtures (for non-Gumbel EV distributions). A third interpretation is through a Peaks over Thresholds model with a positive stable intensity. The mixing variables are used as a modeling tool and for better understanding and model checking. We study extreme value analogues of components of variance models, and new time series, spatial, and continuous parameter models for extreme values. The results are applied to data from a pitting corrosion investigation.