Over the last decade, adversarial attack algorithms have revealed instabilities in deep learning tools. These algorithms raise issues regarding safety, reliability and interpretability in artificial intelligence; especially in high risk settings. From a practical perspective, there has been a war of escalation between those developing attack and defence strategies. At a more theoretical level, researchers have also studied bigger picture questions concerning the existence and computability of attacks. Here we give a brief overview of the topic, focusing on aspects that are likely to be of interest to researchers in applied and computational mathematics.