The Lévy-Lorentz gas describes the motion of a particle on the real line in the presence of a random array of scattering points, whose distances between neighboring points are heavy-tailed i.i.d. random variables with finite mean. The motion is a continuous-time, constant-speed interpolation of the simple symmetric random walk on the marked points. In this seminar I review the model and discuss its large fluctuations and resulting transport properties, both annealed and quenched, under the assumption that the tail distribution of the interdistances between scatterers is regularly varying at infinity. In particular, by presenting large deviation estimates and the asymptotics of moments for the particle displacement, I show that the motion is superdiffusive in the annealed framework, whereas a normal diffusive behavior characterizes the motion conditional on a typical realization of the scatterers arrangement.