Grothendieck proved in EGA IV that if any integral scheme of finite type over a locally noetherian scheme X admits a desingularization, then X is quasi-excellent (we will recall the definition), and conjectured that the converse is probably true. In this talk I will show that the conjecture is true for noetherian schemes of characteristic zero. Namely, starting with the resolution of singularities for algebraic varieties, we will prove the resolution of singularities for noetherian quasi-excellent Q-schemes.