The soft-bodied cephalopods including octopus, cuttlefish, and squid are broadly considered to be the most cognitively advanced group of invertebrates. Previous research has demonstrated that these large-brained molluscs possess a suite of cognitive attributes that are comparable to those found in some vertebrates, including highly developed perception, learning, and memory abilities. Cephalopods are also renowned for performing sophisticated feats of flexible behaviour, which have led to claims of complex cognition such as causal reasoning, future planning, and mental attribution. Hypotheses to explain why complex cognition might have emerged in cephalopods suggest that a combination of predation, foraging, and competitive pressures are likely to have driven cognitive complexity in this group of animals. Currently, it is difficult to gauge the extent to which cephalopod behaviours are underpinned by complex cognition because many of the recent claims are largely based on anecdotal evidence. In this review, we provide a general overview of cephalopod cognition with a particular focus on the cognitive attributes that are thought to be prerequisites for more complex cognitive abilities. We then discuss different types of behavioural flexibility exhibited by cephalopods and, using examples from other taxa, highlight that behavioural flexibility could be explained by putatively simpler mechanisms. Consequently, behavioural flexibility should not be used as evidence of complex cognition. Fortunately, the field of comparative cognition centres on designing methods to pinpoint the underlying mechanisms that drive behaviours. To illustrate the utility of the methods developed in comparative cognition research, we provide a series of experimental designs aimed at distinguishing between complex cognition and simpler alternative explanations. Finally, we discuss the advantages of using cephalopods to develop a more comprehensive reconstruction of cognitive evolution.