  A review of fundamentals and physical applications of fractional quantum
mechanics has been presented. Fundamentals cover fractional Schr\"odinger
equation, quantum Riesz fractional derivative, path integral approach to
fractional quantum mechanics, hermiticity of the Hamilton operator, parity
conservation law and the current density. Applications of fractional quantum
mechanics cover dynamics of a free particle, new representation for a free
particle quantum mechanical kernel, infinite potential well, bound state in
{\delta}-potential well, linear potential, fractional Bohr atom and fractional
oscillator. We also review fundamentals of the L\'evy path integral approach to
fractional statistical mechanics.
