In birational geometry, canonical, log canonical, terminal and log terminal singularities play important roles. These singularities are all normalℚ-Gorenstein singularities and each step of the minimal model program is performed inside the category of normal ℚ-Gorenstein singularities. But in turn, from a purely singularity theoretic view point, the normal ℚ-Gorenstein property seems, in some sense, to be an unnecessary restriction for a singularity to be considered as a good singularity, because there are many “good” singularities without normal ℚ-Gorenstein property (for example, the cone over the Segre embedding ℙ1 x ℙ2 → ℙ5).