Extreme values of real phenomena are events that occur with low frequency, but can have a large impact on real life. These are, in many practical problems, high dimensional by nature. For this purpose, probabilistic models and statistical methods are in high demand. We describe an approach for deriving multivariate extreme value models and we illustrate the main features of some flexible extremal dependence models. We compare them by showing their utility with a real data application, in particular analyzing the extremal dependence among several pollutants recorded in the city of Leeds, UK.