This is an expository note on Hodge theory in the context of moduli spaces of polarized smooth projective varieties. We first recall the system of Hodge bundles, or the graded Higgs bundle, associated with a family as introduced by Simpson. Thanks to Griffiths’ computation of the curvature of the Hodge metric, the kernel of the Higgs map is semi-negative. Building on this negativity, Kawamata and Viehweg established the positivity of the direct image sheaves of powers of the relative dualizing sheaf. We then discuss the deformation Higgs bundle of the universal family on the moduli space, introduced by Viehweg and the author. This bundle extends the classical Kodaira-Spencer map by incorporating Kodaira-Spencer maps of higher degrees. There exists a comparison map between the deformation Higgs bundle and the system of Hodge bundles associated with an auxiliary family over the moduli space, twisted with an anti-ample line bundle arising from the Kawamata-Viehweg positivity theorem. Using this comparison, we demonstrate that the kernel of the Higgs map on the deformation Higgs bundle is strictly negative. This strict negativity of the deformation Higgs bundle has several implications for the geometry of moduli spaces. We outline proofs of Campana-Paun’s theorem on Viehweg hyperbolicity of moduli spaces and Deng-Lu-Sun-Zuo’s theorem on the Picard extension theorem. Finally, we pose a question concerning the topological hyperbolicity of moduli spaces, specifically regarding the exponential growth of the fundamental groups of moduli spaces.