Given an elliptic fibration of a K3 surface, one can reglue the fibres of the elliptic fibration differently to obtain different K3 surfaces (in a one dimensional family of  them). The data of such regluing are dictated either by degree twists or by Brauer twists. A similar story exists for other curves on K3 surfaces. My plan for the talk is to tell this story. Moving from curves to 3-folds, I will report on a joint work in progress with Mattei and Shinder where we look at the cubic 3-folds obtained as the hyperplane sections of a fixed smooth cubic 4-fold. The relative intermediate Jacobians of the universal hyperplanes induces a well behaved abelian fibration over the projective space of dimension 5. The total space this time is a hyperKähler manifold of dimension 10 (again a one-dimensional family of them). The regluings of this abelian fibration gives rise to a Brauer type group associated to the K3-type Hodge structure of the cubic 4-fold.