In this thesis, we study the geometry of Teichmuller space of punctured Riemann surfaces.We use L2 Hodge theory to describe the deformation theory for punctured Riemann surfaces,in which we defined Weil-Petersson metric, Hodge metric and Kodaira-Spencer map. Wealso give a new proof of Wolpert's curvature formula by computing the expansion of volumeform and the Kodaira-Spencer map. We use Wolpert's formula to estimate upper bound forvarious curvature tensor. We construct an extension of pluricanonical form and compare itto the expansion of the Kodaira-Spencer map under Hodge metric
