There is now compelling evidence that many organisms have movement patterns that can be described as Lévy walks, or Lévy flights. Lévy movement patterns have been identified in cells, microorganisms, molluscs, insects, reptiles, fish, birds and even human hunter-gatherers. Most research into Lévy walks as models of organism movement patterns has been shaped by the 'Lévy flight foraging hypothesis'. This states that, since Lévy walks can optimize search efficiencies, natural selection should lead to adaptations that select for Lévy walk foraging. However, a growing body of research on generative mechanisms suggests that Lévy walks can arise freely as by-products of otherwise innocuous behaviours; consequently their advantageous properties are purely coincidental. This suggests that the Lévy flight foraging hypothesis should be amended, or even replaced, by a simpler and more general hypothesis. This new hypothesis would state that 'Lévy walks emerge spontaneously and naturally from innate behaviours and innocuous responses to the environment but, if advantageous, then there could be selection against losing them'. The new hypothesis has the virtue of making fewer assumptions and being broader than the original hypothesis; it also encompasses the many examples of suboptimal Lévy patterns that challenge the prevailing paradigm. This does not detract from the Lévy flight foraging hypothesis, in fact, it adds to the theory by providing a stronger and more compelling case for the occurrence of Lévy walks. It dispenses with concerns about the theoretical arguments in support of the Lévy flight foraging hypothesis and so may lead to a wider acceptance of Lévy walks as models of movement pattern data. Furthermore, organisms can approximate Lévy walks by adapting intrinsic behaviour in simple ways; this occurs when Lévy movement patterns are advantageous, but come with an associated cost. These new developments represent a major change in perspective and provide the broadest picture yet of Lévy movement patterns. However, the process of understanding and identifying Lévy movement patterns still has a long way to go, and further reinterpretations and shifts in understanding will occur. In conclusion, Lévy walk research remains exciting precisely because so much remains to be understood, and because, even relatively small studies, are interesting discoveries in their own right.