Abstract - The talk will concern some recent volume and diameter rigidity results obtained in collaboration with Harish Seshadri and Jian Song. It was shown recently by Kewei Zheng that the volume of an $n$-dimensional Kahler manifold with Ricci curvature larger than $n+1$ is bounded above by the volume of the Fubini-Study metric with Ricci curvature $n+1$. Moreover, it was proved by Yuchen Liu (in an appendix to Zheng's paper) that if the Kahler manifold has almost maximal volume, then it is biholomorphic to the complex projective space. We extend this result to prove that the Kahler manifold is also metrically close, in the sense of Gromov-Hausdorff topology, to the complex projective space with the Fubini-Study metric. The second part of the talk will concern diameter rigidity and I will speak about the Kahler analogue of Cheng's maximal diameter theorem from Riemannian geometry. The correct curvature condition in this case turns out to be a lower bound on the bisectional curvature instead of the Ricci curvature.