Appears in collection : Conference on Arithmetic Geometry in honor of Luc Illusie

I will tell about recent developments in resolution of singularities achieved in a series of works with Abramovich and Wlodarczyk – resolution of log varieties, resolution of morphisms and a no-history (or dream) algorithm for resolution of varieties. I will try to especially emphasize the role of logarithmic geometry in these algorithms and in the quest after them.