Fractional quantum mechanics is a recently emerged and rapidly developing field of quantum physics.This is the first monograph on fundamentals and physical applications of fractional quantum mechanics, written by its founder.The fractional Schroedinger equation and the fractional path integral are new fundamental physical concepts introduced and elaborated in the book. The fractional Schroedinger equation is a manifestation of fractional quantum mechanics. The fractional path integral is a new mathematical tool based on integration over Levy flights. The fractional path integral method...