In recent years, fraudsters have begun to use readily accessible digital manipulation techniques in order to carry out face morphing attacks. By submitting a morph image (a 50/50 average of two people’s faces) for inclusion in an official document such as a passport, it might be possible that both people sufficiently resemble the morph that they are each able to use the resulting genuine ID document. Limited research with low-quality morphs has shown that human detection rates were poor but that training methods can improve performance. Here, we investigate human and computer performance with high-quality morphs, comparable with those expected to be used by criminals. Over four experiments, we found that people were highly error-prone when detecting morphs and that training did not produce improvements. In a live matching task, morphs were accepted at levels suggesting they represent a significant concern for security agencies and detection was again error-prone. Finally, we found that a simple computer model outperformed our human participants. Taken together, these results reinforce the idea that advanced computational techniques could prove more reliable than training people when fighting these types of morphing attacks. Our findings have important implications for security authorities worldwide.