An introduction to covariant quantum mechanics This book deals with an original contribution to the hypothetical missing link unifying the two fundamental branches of physics born in the twentieth century General Relativity and Quantum Mechanics. Namely the book is devoted to a review of a covariant approach to Quantum Mechanics along with several improvements and new results with respect to the previous related literature. The first part of the book deals with a covariant formulation of Galilean Classical Mechanics which stands as a suitable background for covariant Quantum Mechanics. The second part deals with an introduction to covariant Quantum Mechanics. Further in order to show how the presented covariant approach works in the framework of standard Classical Mechanics and standard Quantum Mechanics the third part provides a detailed analysis of the standard Galilean space time along with three dynamical classical and quantum examples. The appendix accounts for several non standard mathematical methods widely used in the body of the book. Introduction Spacetime Galileian metric field Galileian gravitational field Galileian electromagnetic field Joined spacetime connection Classical dynamics Sources of gravitational and electromagnetic fields Fundamental fields of phase space Geometric structures of phase space Hamiltonian formalism Lie algebra of special phase functions Classical symmetries Quantum bundle Galileian upper quantum connection Quantum differentials Quantum dynamics Hydrodynamical picture of QM Quantum symmetries Quantum differential operators Quantum currents and expectation forms Sectional quantum bundle Feynman path integral Conclusions and further developments Examples.