"Evolutionary radiations are key components of the great tree of life, marking points when groups of organisms expand dramatically, and yet the reasons for such expansions are debated: are they driven by new ‘key’ adaptations that make their possessors more successful than their contemporaries, or do they result from major environmental changes that provide new opportunities? Sharks, skates and rays (Neoselachii) are highly successful and have been characterized as perfectly adapted killing machines. They are common in modern oceans, occupying on- and offshore locations as top predators and so maintain the balance and stability of ocean ecosystems and contribute to the regulation of food webs. They have a long evolutionary history (250 Ma) and it seems they replaced precursor shark groups rather rapidly. How did this happen and what were their key adaptations? Although the reasons for their success remain largely unknown, their key novelties include modifications of jaws, sensory systems, physiology and body shape. New numerical comparative phylogenetic methods allow biologists and palaeobiologists to probe questions about the timing of the origins of key evolutionary novelties and to track morphospace evolution. Comparisons of apparent competitors occupying similar ecological niches and food web positions will be made in terms of rates of change in diversity and disparity. The basis for these analyses will be a new inclusive phylogeny of Neoselachii including living and fossil forms for the first time and accordingly will be an important aspect. New numerical techniques will provide key insights into links between diversity and disparity enabling testing of which characters were crucial in driving the initial radiation, the relative importance of different morphological adaptations in their success and the evolutionary steps building towards the perfect predator we see today, but also whether different cartilaginous fish groups might have competed with each other."