Let F be a field. A basic way to study a Brauer class over F is through its splitting fields: field extensions of F over which the class becomes trivial. In this lecture, 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? After explaining the significance of this question, 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. -- This seminar is part of the activities of the Dipartimento di Eccellenza CUP B83C23001390001 and it is funded by the European Union – Next Generation EU.