TITLE: Intuition behind existence of moduli space of stable curves
QUESTION [8 upvotes]: I'm not entirely sure that the title is what I'm looking for. What I'm really asking is for intuition as to why $\bar{\mathcal{M}_g}$ is the compactification of $\mathcal{M}_g$. I'm sure this is covered in the more classic papers (like Deligne and Mumford), but I still find those hard to penetrate.

REPLY [5 votes]: One perspective is geometric. Non-singular closed Riemann surfaces have a unique complete hyperbolic metric by uniformization, with a decomposition into thin and thick regions, where the thin regions are  collar neighborhoods of short geodesics, and the thick regions are compact, with each component having uniformly bounded diameter (depending only on the genus). In the case of a non-compact Riemann surface of finite hyperbolic areaa, one also has cusp components of the thin part.  One also sees that the geometry of a Riemann surface is uniquely determined by the geometry of the thick part, together with which pairs of boundary components bound collar neighborhoods. One may take a limit of a sequence of surfaces with geodesics whose length approaches zero, and the thick parts converge geometrically to a finite area hyperbolic surface, with a pair of cusps for each geodesic whose length has gone to zero. These cusp pairs are naturally identified with noded Riemann surfaces, giving the Deligne-Mumford compactification. The fact that it is compact follows from the compactness of the thick regions in the Gromov-Hausdorff topology. 
It is an interesting result of Masur that the compactification is also given by the metric completion of the Weil-Petersson metric on moduli space.