Unlike most other large mammals, the Soay sheep population of Hirta in the St. Kilda archipelago show persistent oscillations, sometimes increasing or declining by more than 60% in a year. This study explores the causes of these oscillations and their consequences for selection on genetic and phenotypic variation within the population, drawing on studies over the past twenty years of the life-histories and reproductive careers of many sheep. It will be essential reading for vertebrate ecologists, demographers, evolutionary biologists and behavioral ecologists.