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This monograph provides a complete and comprehensive introduction to the theory of long-tailed and subexponential distributions in one dimension. New results are presented in a simple, coherent and systematic way. All the standard properties of such convolutions are then obtained as easy consequences of these results. The book focuses on more theoretical aspects. A discussion of where the areas of applications currently stand in included as is some preliminary mathematical material. Mathematical modelers (for e.g. in finance and environmental science) and statisticians will find this book useful.