The Weil-Petersson metric is not complete metric in general but in the case of abelian varieties and $K3$ surfaces, the Weil-Petersson
metric turns out to be equal to the Bergman metric of the Hermitian symmetric
period domain, hence is in fact complete K\"ahler Einstein metric. Weil and Ahlfors showed that the Weil-Petersson metric is a Kahler metric and later Gang Tian gave a different proof for it also

There is a belief due to Fujiki-Tian saying that a moduli space with
canonical metric (K\"ahler Einstein metric, cscK,...)is likely to be
quasi-projective.

Recently Chi Li http://arxiv.org/pdf/1502.06532.pdf prove that there is a canonical continuous Hermitian metric on
the Tian's CM-line bundle over the proper moduli space $M$ of smoothable Kahler-Einstein Fano varieties. The curvature of this metric is the Weil-Petersson metric. Which Weil-Petersson metric is a Kahler metric

Let $\pi:X \to S$ be a projective family of canonically polarized varieties. Equip
the relative canonical bundle $K_{X/S}$ with the hermitian metric that is induced by the fiberwise K\"ahler-Einstein metrics. The log
Weil-Petersson form is equal, up to a numerical factor, to the fiber integral

$$\omega_{WP}=\int_{X_s}c_1\left(K_{X/S}\right)^{n+1}=\left(\int_{X_s}|A|_{\omega_s}^2\right)ds\wedge d\bar s$$

$A$ represents the Kodaira-Spencer class of the deformation,

There exists a Kahler metric on moduli space of Calabi-Yau varieties
we call it again Weil Petersson metric and can be written as Ricci
curvature of direct image of relative Line bundle.

Take holomorphic fiber space $\pi:X\to B$ and assume $\Psi_y$ be any local non-vanishing holomorphic section of Hermitian line bundle $\pi_*(K_{X/B}^l)$, then the Weil-Petersson (1,1)-form on a small ball $N_r(y)\subset B$ can be written as

$$\omega_{WP}=-\sqrt{-1}\partial_y\bar{\partial_y}\log \left((\sqrt{-1})^{n^2}\int_{X_y}(\Psi_y\wedge \overline{\Psi_y})^{\frac{1}{l}}\right)$$
Note that $\omega_{WP}$ is globally defined on $B$

Fujiki-Schumacher considered the moduli space of Kahler manifolds
admitting constant scalar curvature Kahler (cscK) metrics. They proved
that the natural Weil-Petersson metric is always Kahler by
interpreting it as the Chern curvature of a determinant line bundle
equipped with Quillen metric.