Over recent years there has been an accumulation of evidence from a variety of experimental, theoretical, and field studies that many organisms use a movement strategy approximated by Lévy flights when they are searching for resources. Lévy flights are random movements that can maximize the efficiency of resource searches in uncertain environments. This is a highly significant finding because it suggests that Lévy flights provide a rigorous mathematical basis for separating out evolved, innate behaviors from environmental influences. We discuss recent developments in random-search theory, as well as the many different experimental and data collection initiatives that have investigated search strategies. Methods for trajectory construction and robust data analysis procedures are presented. The key to prediction and understanding does, however, lie in the elucidation of mechanisms underlying the observed patterns. We discuss candidate neurological, olfactory, and learning mechanisms for the emergence of Lévy flight patterns in some organisms, and note that convergence of behaviors along such different evolutionary pathways is not surprising given the energetic efficiencies that Lévy flight movement patterns confer.