Levy flights (LFs), also referred to as Levy motion, stand for a class of non-Gaussian Markovian random processes whose stationary increments are distributed according to a Levy stable distribution originally studied by the French mathematician Paul Pierre Levy. Levy stable laws are important because of three fundamental properties: (i) Similar to the Gaussian law, Levy stable laws form the basin of attraction for sums of random variables. (ii) The probability density functions of Levy stable laws decay in asymptotic power-law form with diverging variance and thus appear naturally in the description of many fluctuation processes with scattering statistics characterized by bursts or large outliers; (iii) LFs are statistically self-affine, a property used for the description of random fractal processes. Here we briefly review the fundamental properties of LFs.