In this survey talk, I will start by explaining the original idea and some of the motivations of Deligne (1984) that there should exist an "irregular Hodge filtration"
sharing some properties with the usual one, on a vector bundle on a curve with connection having irregular singularities at infinity.
I will then describe the present status of the development of this idea as an irregular Hodge theory,
and the various applications to arithmetic, mirror symmetry and confluent hypergeometric differential equations,
that can be made with this theory. The main example developed will be that of the twisted de Rham complex associated to a regular function
on a smooth complex quasi-projective variety, extending the construction of Deligne in his article ‘Hodge II’ (1972).