A quadric bundle is a flat projective morphism every fiber of which is isomorphic to a quadric hypersurface. I will talk about the operations of hyperbolic reduction and hyperbolic extension, inducing an interesting equivalence relation on the set of all quadric bundles over a given base preserving their most important geometric invariants. -- 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.