  Using the identification of the symmetric space
$\mathrm{SL}(n,\mathbb{R})/\mathrm{SO}(n)$ with the Teichm\"uller space of flat
$n$-tori of unit volume, we explore several metrics and compactifications of
these spaces, drawing inspiration both from Teichm\"uller theory and symmetric
spaces. We define and study analogs of the Thurston, Teichm\"uller, and
Weil-Petersson metrics. We show the Teichm\"uller metric is a symmetrization of
the Thurston metric, which is a polyhedral Finsler metric, and the
Weil-Petersson metric is the Riemannian metric of
$\mathrm{SL}(n,\mathbb{R})/\mathrm{SO}(n)$ as a symmetric space. We also
construct a Thurston-type compactification using measured foliations on
$n$-tori, and show that the horofunction compactification with respect to the
Thurston metric is isomorphic to it, as well as to a minimal Satake
compactification.
