Rational curves, rational points and fixed points

There is a strong analogy between continuous sections of a topological fibration, rational points of a variety defined over a non-closed field, and fixed points of the action of a finite group. I will explain this analogy as well as a number of recent results for rational points and fixed points suggested by this analogy: a theorem of de Jong-He-Starr, Yi Zhu's theorem, the Tian-Zong theorem, and work-in-progress of Chenyang Xu and myself for rational points of varieties over global function fields.