This book contains a selection of more than 500 mathematical problems and their solutions from the PhD qualifying examination papers of more than ten famous American universities. The problems cover six aspects of graduate school mathematics: Algebra, Differential Geometry, Topology, Real Analysis, Complex Analysis and Partial Differential Equations. The depth of knowledge involved is not beyond the contents of the textbooks for graduate students, while solution of the problems requires deep understanding of the mathematical principles and skilled techniques. For students this book is a valuable complement to textbooks; for lecturers teaching graduate school mathematics, a helpful reference.