In this talk, I will discuss the rigidity problem for compact submanifolds under sectional curvature pinching condition. Inspired by the well-known Yau rigidity theorem, we first study the geometric rigidity of compact submanifolds with parallel mean curvature in a space form. Then, we investigate the sphere theorems for general submanifolds in a space form, and obtain some sharp results.  At last, we discuss the rigidity of compact submanifolds with parallel mean curvature in pinched manifolds, and prove some classification theorems. This is a joint work with Prof. Hongwei Xu and Dr. Li Lei.