The resolution of singular foliations on analytic manifolds and algebraic varieties is a notoriously challenging problem, with only a few known partial results. In characteristic zero, we construct principalization of ideals on smooth foliated varieties. As an application, we prove the desingularization of Darboux totally integrable foliations in arbitrary dimensions, including both rational and meromorphic Darboux foliations. Additionally, we show how to transform a generically transverse section into a fully transverse one. Our approach relies on torus actions and uses weighted cobordant blow-ups, and is closely related to the analogous method of resolution of singularities of varieties.This is joint work with Abramovich, Belotto, and Temkin.[-]