  This is a survey of our recent results on the geometry of moduli spaces and
Teichmuller spaces of Riemann surfaces appeared in math.DG/0403068 and
math.DG/0409220. We introduce new metrics on the moduli and the Teichmuller
spaces of Riemann surfaces with very good properties, study their curvatures
and boundary behaviors in great detail. Based on the careful analysis of these
new metrics, we have a good understanding of the Kahler-Einstein metric from
which we prove that the logarithmic cotangent bundle of the moduli space is
stable. Another corolary is a proof of the equivalences of all of the known
classical complete metrics to the new metrics, in particular Yau's conjectures
in the early 80s on the equivalences of the Kahler-Einstein metric to the
Teichmuller and the Bergman metric.
