Biometric presentation attack detection (PAD) is gaining increasing attention. Users of mobile devices find it more convenient to unlock their smart applications with finger, face, or iris recognition instead of passwords. In this study, the authors survey the approaches presented in the recent literature to detect face and iris presentation attacks. Specifically, they investigate the effectiveness of fine-tuning very deep convolutional neural networks to the task of face and iris antispoofing. They compare two different fine-tuning approaches on six publicly available benchmark datasets. Results show the effectiveness of these deep models in learning discriminative features that can tell apart real from fake biometric images with a very low error rate. Cross-dataset evaluation on face PAD showed better generalisation than state-of-the-art. They also performed cross-dataset testing on iris PAD datasets in terms of equal error rate, which was not reported in the literature before. Additionally, they propose the use of a single deep network trained to detect both face and iris attacks. They have not noticed accuracy degradation compared to networks trained for only one biometric separately. Finally, they analysed the learned features by the network, in correlation with the image frequency components, to justify its prediction decision.