We'll be talking about recent results on the problem of splitting Brauer classes by torsors for genus one curves. In its geometric form the question to be asked is: does every Severi--Brauer variety contain a smooth and projective genus one curve? Algebraically, this question is related to the existence of certain finite Galois modules inside the linear algebraic automorphism group of the Severi--Brauer variety. Our goal will be to motivate why this is an intuitive and interesting question, giving some new results along the way.