This talk will discuss Chern--Schwartz--MacPherson (CSM) classes over classical flag varieties. After a brief introduction to CSM classes, we will explain our main theorem --- an equivariant CSM Pieri and Murnaghan--Nakayama rule which includes as special cases many previously known formulas for CSM classes or Schubert classes. If time permitted, we will discuss related conjectures on the positivities of CSM classes. This talk is based on the joint work with Neil J.Y. Fan and Peter L. Guo.