First, it introduces the derivation and background of quantum hydrodynamic equations. In the second chapter, global existence of the weak solutions to the compressible quantum Navier-Stokes equations is introduced. Existence of the finite energy weak solution to the inviscid quantum hydrodynamic equations is considered in Chapter 3. Chapter 4 introduces the existence of weak solutions to the non-isentropic quantum Navier-Strokes equations with cold pressure. The initial boundary value problem of the compressible quantum Navier-Strokes equations is considered in Chapter 5, and the asymptotic limit to the bipolar quantum hydrodynamic equations is introduced in the final chapter. This book can be used a basic reference book for nonlinear mathematics, physics and engineering workers. We hope the publishing of this book will help the readers to sort out issues that are not clarified in previous books and literatures, enabling the reader to carry out basic research by referring to the relevant references when they are interested in specific aspect of the problem.