In this talk, we will discuss Conjecture O and Gamma conjecture I for Fano manifolds in more details. We will review the current approaches to both conjectures in various concrete examples. We will also introduce some relevant conjectures on the eigenvalues of linear operators on the quantum cohomology of a Fano manifold, induced by the quantum multiplications. This is the third one of the series of talks on Gamma conjectures, and the context of this talk will be self-contained.