This paper introduces a new approach to fractional quantum mechanics (fQM) and fractional statistical mechanics (fSM) based on path integrals over measures induced by Lévy flights. The Lévy flights generalize Brownian motion, which is described by the Wiener process and the diffusion equation. The Lévy process is characterized by stable probability distributions, which are generalizations of the Gaussian distribution. Lévy flights are used to model various physical processes, including anomalous diffusion, turbulence, and financial dynamics. The paper develops the foundations of fQM and fSM using a new path integral formulation. The fractional quantum mechanics is based on the Laskin path integral, which generalizes the Feynman path integral. The Laskin path integral measure is defined using the Lévy function, which is expressed in terms of Fox's H functions. The fractional free particle propagator is derived and shown to reduce to the Feynman propagator when α = 2. The energy of the fractional quantum mechanical particle is given by E_p = D_α |p|^{α}. In statistical mechanics, the fractional density matrix is derived, which generalizes the Wiener measure. The fractional density matrix for a free particle is shown to reduce to the well-known density matrix for a classical particle when α = 2. The fractional density matrix obeys a fractional differential equation. The paper concludes that the new path integral approach to fQM and fSM is based on the stochastic process of Lévy flights. This approach provides a generalization of the Feynman and Wiener path integrals and offers a new framework for studying quantum and statistical systems with non-Gaussian behavior. The paper suggests that further research is needed to analyze the physical systems governed by the developed fQM and fSM.This paper introduces a new approach to fractional quantum mechanics (fQM) and fractional statistical mechanics (fSM) based on path integrals over measures induced by Lévy flights. The Lévy flights generalize Brownian motion, which is described by the Wiener process and the diffusion equation. The Lévy process is characterized by stable probability distributions, which are generalizations of the Gaussian distribution. Lévy flights are used to model various physical processes, including anomalous diffusion, turbulence, and financial dynamics. The paper develops the foundations of fQM and fSM using a new path integral formulation. The fractional quantum mechanics is based on the Laskin path integral, which generalizes the Feynman path integral. The Laskin path integral measure is defined using the Lévy function, which is expressed in terms of Fox's H functions. The fractional free particle propagator is derived and shown to reduce to the Feynman propagator when α = 2. The energy of the fractional quantum mechanical particle is given by E_p = D_α |p|^{α}. In statistical mechanics, the fractional density matrix is derived, which generalizes the Wiener measure. The fractional density matrix for a free particle is shown to reduce to the well-known density matrix for a classical particle when α = 2. The fractional density matrix obeys a fractional differential equation. The paper concludes that the new path integral approach to fQM and fSM is based on the stochastic process of Lévy flights. This approach provides a generalization of the Feynman and Wiener path integrals and offers a new framework for studying quantum and statistical systems with non-Gaussian behavior. The paper suggests that further research is needed to analyze the physical systems governed by the developed fQM and fSM.