Fractional quantum mechanics Fractional quantum mechanics is a recently developed natural generalization of quantum mechanics. This book summarizes fundamentals and applications of fractional quantum mechanics. The book presents the fractional Schrödinger equation which is a manifestation of fractional quantum mechanics. A new theoretical tool fractional path integral the path integral over Lévy flights has been presented. The fractional path integral method enhances the well known Feynman path integral framework. The book covers related topics on time fractional quantum mechanics fractional classical mechanics and the [alpha] stable Lévy random process. The book is helpful as a monograph and a handbook for theorists pure and applied mathematicians solid state physicists and chemists working with the Schrödinger equation the path integral technique and applications of fractional calculus in different research areas What is fractional quantum mechanics Fractals Fractional Schrödinger equation Time independent fractional Schrödinger equation Fractional uncertainty relation Path integral over Lévy flights A free particle quantum kernel Transforms of a free particle kernel Fractional oscillator Some analytically solvable models of fractional quantum mechanics Fractional non linear quantum dynamics Time fractional quantum mechanics Applications of time fractional quantum mechanics Fractional statistical mechanics Fractional classical mechanics Fractional dynamics in polar coordinate system