Media forensics has attracted a lot of attention in the last years in part due to the increasing concerns around DeepFakes. Since the initial DeepFake databases from the 1st generation such as UADFV and FaceForensics++ up to the latest databases of the 2nd generation such as Celeb-DF and DFDC, many visual improvements have been carried out, making fake videos almost indistinguishable to the human eye. This study provides an exhaustive analysis of both 1st and 2nd DeepFake generations in terms of facial regions and fake detection performance. Two different methods are considered in our experimental framework: i) the traditional one followed in the literature and based on selecting the entire face as input to the fake detection system, and ii) a novel approach based on the selection of specific facial regions as input to the fake detection system. Among all the findings resulting from our experiments, we highlight the poor fake detection results achieved even by the strongest state-of-the-art fake detectors in the latest DeepFake databases of the 2nd generation, with Equal Error Rate results ranging from 15% to 30%. These results remark the necessity of further research to develop more sophisticated fake detectors.