I will give a general introduction to the study of the Hodge filtration on local cohomology sheaves associated to closed subschemes of smooth complex varieties, using techniques from both D-module theory and birational geometry. In the case of hypersurfaces, this is essentially the theory of Hodge ideals, which I will recall. This study has applications to various topics, like local vanishing, local cohomological dimension, the Du Bois complex, minimal exponents of singularities, etc. I will discuss a few, and more will appear in M. Mustaja's lecture.[-]