Abstract: Let F be a field. A basic way to study a Brauer class over F is through its splitting fields, that is, field extensions of F over which the class becomes trivial. I will discuss a question of Clark and Saltman: is every Brauer class over F split by a smooth projective genus one curve over F? Equivalently, does every Severi-Brauer variety X/F admit a morphism C -> X, with C a smooth projective curve of genus one? I will present joint work with Zinovy Reichstein showing that the answer is no in general. I will then discuss joint work with Ben Antieau and Asher Auel showing that, in contrast, the answer is yes over number fields.