  We introduce a category of possibly irregular holonomic D-modules which can
be endowed in a canonical way with an irregular Hodge filtration. Mixed Hodge
modules with their Hodge filtration naturally belong to this category, as well
as their twist by $\exp\varphi$ for any meromorphic function $\varphi$. This
category is stable by various standard functors, which produce many more
filtered objects. The irregular Hodge filtration satisfies the
$E_1$-degeneration property by a projective morphism. This generalizes some
results proved by Esnault-Sabbah-Yu arxiv:1302.4537 and Sabbah-Yu
arxiv:1406.1339. We also show that those rigid irreducible holonomic D-modules
on the complex projective line whose local formal monodromies have eigenvalues
of absolute value one, are equipped with such an irregular Hodge filtration in
a canonical way, up to a shift of the filtration. In a chapter written jointly
with Jeng-Daw~Yu, we make explicit the case of irregular mixed Hodge
structures, for which we prove in particular a Thom-Sebastiani formula.
