  A brief history of the investigation of the Weil-Petersson curvature and a
summary of Teichm\"{u}ller theory are provided. A report is presented on the
program to describe an intrinsic geometry with the Weil-Petersson metric and
geodesic-length functions. Formulas for the metric, covariant derivative and
formulas for the curvature tensor are presented. A discussion of methods is
included. Recent and new applications are sketched, including results from the
work of Liu-Sun-Yau, an examination of the Yamada model metric and a
description of Jacobi fields along geodesics to the boundary.
