Abstract: Comparison geometry is a significant topic in metric geometry and geometric analysis. Gromov proposes to study comparison theorems related to scalar curvature and mean curvature for manifolds with boundary. In this talk, I will summarize recent developments on this topic, utilizing tools from index theory and geometric analysis. I will also talk about our recent work, joint with Zhichao Wang and Bo Zhu, where we prove a scalar-mean rigidity theorem using capillary mu-bubbles.