  This paper gives a self-contained introduction to the Hilbert projective
metric $\mathcal{H}$ and its fundamental properties, with a particular focus on
the space of probability measures. We start by defining the Hilbert
pseudo-metric on convex cones, focusing mainly on dual formulations of
$\mathcal{H}$ . We show that linear operators on convex cones contract in the
distance given by the hyperbolic tangent of $\mathcal{H}$, which in particular
implies Birkhoff's classical contraction result for $\mathcal{H}$. Turning to
spaces of probability measures, where $\mathcal{H}$ is a metric, we analyse the
dual formulation of $\mathcal{H}$ in the general setting, and explore the
geometry of the probability simplex under $\mathcal{H}$ in the special case of
discrete probability measures. Throughout, we compare $\mathcal{H}$ with other
distances between probability measures. In particular, we show how convergence
in $\mathcal{H}$ implies convergence in total variation, $p$-Wasserstein
distance, and any $f$-divergence. Furthermore, we derive a novel sharp bound
for the total variation between two probability measures in terms of their
Hilbert distance.
