Let A be a basic ?nite dimensional algebra over an algebraically closed ?eld k and n be a positive integer. A is said to be an n-Auslander algebra (resp. n-Auslander-Gorenstein algebra) if it satis?es gl.d A ≤ n + 1 ≤ dom.d A (resp. id A ≤ n + 1 ≤ dom.d A). In particular, 1-Auslander algebra is the classical Auslander algebra. In this talk, we give some homological properties of Auslander algebras, n-Auslander algebras and n-Auslander-Gorenstein algebras. As applications, we obtain some new characterization for these three kinds of algebras.