Extreme value theory (EVT) is commonly applied in several fields, such as finance, hydrology and environmental modeling. It is extensively developed in the univariate setting. A number of studies have focused on the extension of EVT to the multivariate context. However, most of these studies are based on a direct extension of univariate extremes. In the present paper, we present a procedure to identify the extremes in a multivariate sample. The present procedure is based on the statistical notion of depth function combined with the orientation of the observations. The extreme identification itself is important and it can also serve as basis for the modeling and the asymptotic studies. The proposed procedure is also employed to detect peaks-over-thresholds in the multivariate setting. This method is general and includes several special cases. Furthermore, it is flexible and can be applied to several situations depending on the degree of extreme event risk. The procedure is mainly motivated by application considerations. A simulation study is carried out to evaluate the performance of the procedure. An application, based on air quality data, is presented to show the various elements of the procedure. The procedure is also shown to be useful in other statistical areas.