Abstract: The real-space approach has been proved as a powerful method to study topological states: both the classification of topological states and that of their anomalous surfaces can be obtained in an intuitive, straightforward way. I will talk about new progress in topological materials which is inspired by, the real-space approach. Specifically, a new kind of bulk-boundary correspondence for the so-called fragile topology and the anomalous hinge modes of the so-called higher-order topological insulator, are developed. The theory is tested in an experimental setting. The real space approach also provides a new perspective for the localization problem in disordered topological states.