A C*-algebra is said to be Z-stable if its tensor product with the Jiang-Su algebra is isomorphic to itself, and the Z-stability plays a central role in the recent classification theorem for C*-algebras. In the talk, let us examine this property and beyond: In the first part of the talk, let us consider Villadsen algebras, a class of AH algebras which are not Z-stable, and show that, if the seed space is a given contractible compact metric space, they are indeed classified by the conventional Elliott invariant together with the radius of comparison. In the second part of the talk, let us return to the property of Z-stability and study its relation to mean dimension and the small boundary property. The talk is based on a joint work with George Elliott and Chunguang Li, and a joint work with George Elliott.