Search processes play an important role in physical, chemical and biological systems, including, for example, the encounter of two molecules to perform a chemical reaction, proteins searching megabases of DNA for specific binding sites, and in animal foraging. Specialised random walks known as Lévy flights have been shown in simulations to be an efficient search strategy under certain conditions to locate new resource patches by combining local search with longer excursions to new areas. Theoretical results present the hypothesis that Lévy flights optimise random searches, therefore biological organisms must have evolved to exploit Lévy flights (the so-called Lévy flight foraging hypothesis). Much recent research has tested the Lévy flight foraging hypothesis with empirical data, apparently finding Lévy search patterns in diverse species, from bacteria to humans. Is Lévy flight searching, therefore, a universal law in ecology? Do these empirical studies stand up to closer scrutiny? This seminar will describe the scientific background to this topic, identify key points of general interest in the debate, and will present empirical results from the movement analysis of electronically tagged marine predators. Recent studies will be presented for sharks, tuna, swordfish and turtles, that offer themselves as particularly good models for testing the Lévy flight foraging hypothesis.