  Let $\widetilde{\cal M}_{g,n}$, for $2g-2+n>0$, be the moduli stack of
$n$-pointed, genus $g$, stable complex curves of compact type. Various
characterizations and properties are obtained of both the algebraic and
topological fundamental groups of the stack $\widetilde{\cal M}_{g,n}$.
  Let $\Gamma_{g,n}$, for $2g-2+n>0$, be the Teichm\"uller group associated
with a compact Riemann surface of genus $g$ with $n$ points removed $S_{g,n}$,
i.e. the group of homotopy classes of diffeomorphisms of $S_{g,n}$ which
preserve the orientation of $S_{g,n}$ and a given order of its punctures. Let
$K_{g,n}$ be the normal subgroup of $\Gamma_{g,n}$ generated by Dehn twists
along separating circles on $S_{g,n}$. As a first application of the above
theory, a characterization of $K_{g,n}$ is given for all $n\geq 0$ (for
$n=0,1$, this was done by Johnson). Let then ${\cal T}_{g,n}$ be the Torelli
group, i.e. the kernel of the natural representation $\Gamma_{g,n}\ra
Sp_{2g}(Z)$. The abelianization of ${\cal T}_{g,n}$ is determined for all
$g\geq 1$ and $n\geq 1$, thus completing classical results by Johnson and Mess.
