The central focus of this thesis is on the index of genus one curves. We prove existence of such curves with prescribed index over fields finitely generated over $bbF_p(t)$. The proof is by induction on the transcendence degree. This generalizes -- and uses as the base case of an inductive argument -- an older result on the number field case. There is a separate base case in every positive characteristic $p$, and these use work on the conjecture of Birch and Swinnerton-Dyer over function fields.