Slow dimension growth condition is one of essentials in the classification of simple amenable C*-algebras, and it implies that the algebra is regular for the purpose of the classification, i.e., the algebra absorbs the Jiang-Su algebra Z tensorially. Consider a minimal homeomorphism with mean dimension zero, then the corresponding C*-algebra is shown to have slow dimension growth, and hence is covered by the recent progress of the classification program. In particular, this includes the C*-algebra of any uniquely ergodic system. The talk is based on a joint work with George A. Elliott.