  In this paper, we construct spines, i.e., $\Mod_g$-equivariant deformation
retracts, of the Teichm\"uller space $\T_g$ of compact Riemann surfaces of
genus $g$. Specifically, we define a $\Mod_g$-stable subspace $S$ of positive
codimension and construct an intrinsic $\Mod_g$-equivariant deformation
retraction from $\T_g$ to $S$. As an essential part of the proof, we construct
a canonical $\Mod_g$-deformation retraction of the Teichm\"uller space $\T_g$
to its thick part $\T_g(\varepsilon)$ when $\varepsilon$ is sufficiently small.
These equivariant deformation retracts of $\T_g$ give cocompact models of the
universal space $\underline{E}\Mod_g$ for proper actions of the mapping class
group $\Mod_g$. These deformation retractions of $\T_g$ are motivated by the
well-rounded deformation retraction of the space of lattices in $\R^n$. We also
include a summary of results and difficulties of an unpublished paper of
Thurston on a potential spine of the Teichm\"uller space.
