The theory of mixed Hodge modules (MHM) is a significant generalization  of the classical Hodge theory. It unifies the classical theory and the theories of variations of Hodge structure, mixed Hodge structures and variations of mixed Hodge structure, to allow us a functorial treatment.
For example, for a polynomial mapping f from C^n to C, we consider the ``pushforward" (of some degree) of a constant sheaf on C^n by f. Then, we get a perverse sheaf (= a generalization of local systems or monodromy representation) and a regular holonomic D-module (= a generalization of a linear differential equation) called the Gauss-Manin system.
By the functorial property of MHMs, the pair of these objects is equipped with a Hodge and weight filtrations, which define a MHM.

While MHMs (or variations of mixed Hodge structure) on the complex line C are classical and basic, and their general properties have been studied, they are complicated objects and not yet fully understood. However, we can completely describe a MHM when it satisfies the condition called monodromic. As an application of it, we give a natural definition of the Fourier-Laplace transform of a monodromic MHM and also a concrete description of the irregular Hodge filtration of it.

In the first half of this talk, I will review ``the Hodge theory before MHM" (including a brief explanation of Hodge structures, variations of Hodge structure, mixed Hodge structures and D-modules) and "the theory of MHMs". In the second half, I will introduce my results on monodromic mixed Hodge modules.