  After a short excursion from discovery of Brownian motion to the Richardson
"law of four thirds" in turbulent diffusion, the article introduces the
L\'{e}vy flight superdiffusion as a self-similar L\'{e}vy process. The
condition of self-similarity converts the infinitely divisible characteristic
function of the L\'{e}vy process into a stable characteristic function of the
L\'{e}vy motion. The L\'{e}vy motion generalizes the Brownian motion on the
base of the $\alpha$-stable distributions theory and fractional order
derivatives. The further development of the idea lies on the generalization of
the Langevin equation with a non-Gaussian white noise source and the use of
functional approach. This leads to the Kolmogorov's equation for arbitrary
Markovian processes. As particular case we obtain the fractional Fokker-Planck
equation for L\'{e}vy flights. Some results concerning stationary probability
distributions of L\'{e}vy motion in symmetric smooth monostable potentials, and
a general expression to calculate the nonlinear relaxation time in barrier
crossing problems are derived. Finally we discuss results on the same
characteristics and barrier crossing problems with L\'{e}vy flights, recently
obtained with different approaches.
