  A L\'evy random medium, in a given space, is a random point process where the
distances between points, a.k.a. targets, are long-tailed. Random walks
visiting the targets of a L\'evy random medium have been used to model many
(physical, ecological, social) phenomena that exhibit superdiffusion as the
result of interactions between an agent and a sparse, complex environment. In
this note we consider the simplest non-trivial L\'evy random medium, a sequence
of points in the real line with i.i.d. long-tailed distances between
consecutive targets. A popular example of a continuous-time random walk in this
medium is the so-called L\'evy-Lorentz gas. We give an account of a number of
recent theorems on generalizations and variations of such model, in discrete
and continuous time.
