Kodaira vanishing is the most famous vanishing theorem in algebraic geometry. After reviewing Kodaira-Nakano vanishing, I will introduce its generalization to mixed Hodge modules due to Saito and a recent Kawamata-Viehweg type vanishing for pure Hodge modules. In order to understand Q-divisors with normal crossing support, I will introduce log-D modules and construct a special kind of log-D modules, Multi-indexed Deligne extensions from C-local systems. After this, I will deduce a Kawamata-Viehweg type vanishing for log-variations of Hodge structures (log-VHS). If time allowed, I will also mention how to construct multiplier subsheaves for Hodge modules based on log-VHS which generalizes multiplier ideal sheaves. Speaker(s): Lei Wu (Northwestern University)