Approximations of C∗-algebras and the ideal property

We introduce several classes of C∗-algebras (using for this local approximations by “nice” C∗-algebras), that generalize the AH algebras, among others. We initiate their study, proving mainly results about the ideal property, but also about the ideals generated by their projections, the real rank zero, the weak projection property, minimal tensor products, extensions, quasidiagonal extensions, ideal structure, the largest ideal with the ideal property and short exact sequences. Some of the previous results of the second named author are generalized.