  We study the geometry of the Thurston metric on the Teichm\"uller space
$\mathcal{T}(S)$ of hyperbolic structures on a surface $S$. Some of our results
on the coarse geometry of this metric apply to arbitrary surfaces $S$ of finite
type; however, we focus particular attention on the case where the surface is a
once-punctured torus, $S_{1,1}$. In that case, our results provide a detailed
picture of the infinitesimal, local, and global behavior of the geodesics of
the Thurston metric, as well as an analogue of Royden's theorem.
