  This is a survey paper on the topic of Weil-Petersson geometry of Teichmuller
spaces. Even though historically the subject has been developed as a branch of
complex analysis, the treatment here is from the view-point of differential
geometry, much influenced by the works of Eells, Earle, Fischer, Tromba and
Wolpert over the several decades. The Weil-Petersson geometry has negative
sectional curvature, and the main theme of this article is to exploit the
convexity of the Weil-Petersson distance function induced by the negativity of
the curvature, from Riemannian geometric approaches as well as those of the
CAT(0) geometry and Coxeter theory.
