We introduce and study new complete Kahler metrics on the moduli and the Teichmuller spaces of Riemann surfaces, the Ricci and the perturbed Ricci metric. They are asymptotically equivalent to the Poincare metric. The perturbed Ricci metric has bounded negative sectional and Ricci curvatures. As corollaries we prove the equivalence of these new metrics to several classical metrics such as the Kahler-Einstein metrics, proving a conjecture of Yau in the early 80s. Other consequences will also be discussed. This is joint work with K. Liu and S.-T. Yau.