  We consider a version of continuum long-range percolation on finite boxes of
$\mathbb{R}^d$ in which the vertex set is given by the points of a Poisson
point process and each pair of two vertices at distance $r$ is connected with
probability proportional to $r^{-s}$ for a certain constant $s$. We explore the
graph-theoretical distance in this model. The aim of this paper is to show that
this random graph model undergoes phase transitions at values $s=d$ and $s=2d$
in analogy to classical long-range percolation on $\mathbb{Z}^d$, by using
techniques which are based on an analysis of the underlying Poisson point
process.
