I will present a comprehensive framework for the resolution of singularities, highlighting its applications to foliations. The core of this approach relies on blow-ups with weighted centers - a natural and powerful tool that utilizes simple geometric invariants derived from the weighted normal cone to approximate singularities and guide resolution through successive improvements.

While this machinery is fundamentally rooted in the resolution of algebraic varieties in characteristic zero (and, in some cases, positive characteristic), this framework extends naturally to the setting of foliated varieties, and some classes of foliations. The method integrates torus actions and Rees algebras, working within smooth ambient spaces with simple normal crossings divisors to achieve embedded resolution, principalization of ideals, and the resolution of foliations.

These results stem from a series of joint projects with D. Abramovich and M. Temkin, as well as joint work with Abramovich, Belotto, and Temkin.