Rational simple connectedness is an algebro-geometric analogue of simple connectedness replacing continuous maps from the unit interval
with morphisms from the projective line. Using this notion, de Jong, He and I proved the "split case" of Serre's "Conjecture II" for function fields of surfaces over an algebraically closed field: every principal bundle over a surface for a semisimple, simply connected algebraic group has a rational section. In particular, the split case implies the E8 case. Combined work of Merkurjev-Suslin, Ojanguren-Parimala, Colliot-Th\'el\`ene - Gille - Parimala and Gille reduced the general case of Serre's Conjecture II for function fields to the E8 case. Thus 
Serre's Conjecture II is now settled for function fields of surfaces over an 
algebraically closed field. I will explain rational simple 
connectedness and how it applies to Serre's Conjecture II.