The book covers the theory of Quantum Gravitation from the point of view of Feynman path integrals. These provide a manifestly covariant approach in which fundamental quantum aspects of the theory such as radiative corrections and the renormalization group can be systematically and consistently addressed. The path integral method is suitable for both perturbative as well as non-perturbative studies, and is known to already provide a framework of choice for the theoretical investigation of non-abelian gauge theories, the basis for three of the four known fundamental forces in nature. The book thus provides a coherent outline of the present status of the theory gravity based on Feynman's formulation, with an emphasis on quantitative results. Topics will be organized in such a way that the correspondence to similar methods and results in modern gauge theories will become apparent. Covariant perturbation theory will be developed using the full machinery of Feynman rules, gauge fixing, background methods and ghosts. The renormalization group for gravity and the existence of non-trivial ultraviolet fixed points will be developed, stressing a close correspondence with well understood statistical field theory models. Later the lattice formulation of gravity will be presented as the essential tool in understanding key features of the non-perturbative vacuum of quantum gravity. At the end the book will discuss contemporary issues in quantum cosmology such as scale dependent gravitational constants and quantum effects in the early universe. When feasible he above topics will be presented in a self-contained way, stressing the basic physical ideas, developing simple examples and exploiting analogies where suitable. Simplicity in the presentation will take precedence over excessive mathematical rigor, and many illustrations should help stimulate the reader's interest, and guide his understanding. The book should provide a self-contained, yet comprehensive, introduction to quantum gravitation based on the traditional, well tested covariant approach that forms the basis for a modern treatment of gauge theories. It is intended to be suitable as a second-year book for graduate students with an interest in quantum cosmology, quantum gravity and its relation to non-abelian gauge theories and modern statistical field theory. TOC:Continuum Formulation.- Feynman Path Integral Formulation.- Gravity in 2+e Dimensions.- Hamiltonian and Wheeler-DeWitt Equation.- Semiclassical Gravity.- Lattice Regularized Quantum Gravity.- Analytical Lattice Expansion Methods.- Numerical Studies.- Scale Dependent Gravitational Couplings.