  This article provides an overview of recent work on descriptions and
properties of the convex minorant of random walks and L\'evy processes which
summarize and extend the literature on these subjects.
  The results surveyed include point process descriptions of the convex
minorant of random walks and L\'evy processes on a fixed finite interval, up to
an independent exponential time, and in the infinite horizon case. These
descriptions follow from the invariance of these processes under an adequate
path transformation. In the case of Brownian motion, we note how further
special properties of this process, including time-inversion, imply a
sequential description for the convex minorant of the Brownian meander.
