This book grew out of a set of notes that I supplied to the audience of a series of lectures on "The Thomas-Fermi Method in Atomic Physics and Its Refinements" delivered at the University of Munich in 1985. Standard textbook material played a minor role during these lectures the emphasis was on the novel approach developed by Professor Julian Schwinger and myself, beginning about eight years ago. As a consequence, this book is the first complete, detailed step-by-step presentation of our ideas and their implications.