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.