Let Y be a Calabi—Yau variety. The Morrison Cone Conjecture is a fundamental conjecture in Algebraic Geometry on the geometry of the nef cone and the movable cone of Y: while these cones are usually not rational polyhedral, the cone conjecture predicts that the action of Aut(Y) on Nef(Y) admits a rational polyhedral fundamental domain, and that the action of Bir(Y) on Mov(Y) admits a rational polyhedral fundamental domain. Even though the conjecture has been settled in special cases, it is still wide open in dimension at least 3. We prove that if the cone conjecture holds for a smooth Calabi-Yau threefold Y, then it also holds for any smooth deformation of Y.  If time permits, I will explain how to generalize some of the ideas to the case of log Calabi-Yau varieties.