The curve complex is a simplicial complex encoding the intersection patterns of simple closed curves on a surface. There is a natural mapping class group action on the complex and we will discuss different rigidity results on the action. In particular, we will look at the finite rigidity of the curve complex proved by Aramayona-Leininger, i.e. one can always recover a mapping class from the translation of a finite subcomplex. Thus the curve complex serves as a good model to study the mapping class group: there is not much flexibility in the complex other than the automorphisms induced by the mapping classes. Similar rigidity phenomenon occurs when one considers other natural variations of the curve complex, such as the separating curve complex. However, such complexes are not as well-studied as the curve complex. One question is how the whole story of the curve complex looks like on other complexes. The talk is based on joint work with Bena Tshishiku.