  We develop a theory of Bridgeland stability conditions and moduli spaces of
semistable objects for a family of varieties. Our approach is based on and
generalizes previous work by Abramovich-Polishchuk, Kuznetsov, Lieblich, and
Piyaratne-Toda. Our notion includes openness of stability, semistable
reduction, a support property uniformly across the family, and boundedness of
semistable objects. We show that such a structure exists whenever stability
conditions are known to exist on the fibers.
  Our main application is the generalization of Mukai's theory for moduli
spaces of semistable sheaves on K3 surfaces to moduli spaces of Bridgeland
semistable objects in the Kuznetsov component associated to a cubic fourfold.
This leads to the extension of theorems by Addington-Thomas and Huybrechts on
the derived category of special cubic fourfolds, to a new proof of the integral
Hodge conjecture, and to the construction of an infinite series of unirational
locally complete families of polarized hyperk\"ahler manifolds of K3 type.
  Other applications include the deformation-invariance of Donaldson-Thomas
invariants counting Bridgeland stable objects on Calabi-Yau threefolds, and a
method for constructing stability conditions on threefolds via degeneration.
