In the early 1960's Serre formulated two conjectures about Galois cohomology. The first was proved by Steinberg shortly thereafter, but the second remains open. I will discuss the proof of Serre's Conjecture II in the "geometric case": every principal homogeneous space for a bundle of simply connected, semisimple groups over a surface has a rational section. Due to the work of many people — Merkurjev and Suslin, E. Bayer and Parimala, Chernousov, Gille—the geometric case further reduces to the "split, geometric case" i.e., the bundle of groups is constant. And this case was proved by de Jong, X. He and myself using "rational simple connectedness." No background in Galois cohomology or rational connectedness will be assumed.