Abstract: Generalized Donaldson–Thomas invariants defined by Joyce and Song are rational numbers which “count” both τ-stable and τ-semistable coherent sheaves with Chern character α on a Calabi–Yau 3-fold X, where τ denotes Gieseker stability for some ample line bundle on X. These invariants are defined for all classes α, and are equal to the classical DT defined by Thomas when it is defined. They are unchanged under deformations of X, and transform by a wall-crossing formula under change of stability condition τ. Joyce and Song use gauge theory and transcendental complex analytic methods, so that the theory of generalized Donaldson–Thomas invariants is valid only in the complex case. This also forces them to put constraints on the Calabi–Yau 3-fold they can define generalized ….