The solution of the problem of defining and existence of a dual continuous Banach bundle (CBB) for the case of ample bundles over extremally disconnected compact spaces is essentially based on specific properties of ample bundles and extremally disconnected compact spaces and, thus, cannot be extended to a wider class of bundles. The natural intention to extend the domain of applications for the duality theory leads to the problem of constructing a dual CBB for an arbitrary Banach bundle over an arbitrary topological space. The study of this problem is the main subject of the present chapter, where, in particular, a definition of a dual bundle is presented, with the above-formulated requirements fulfilled, and a number of necessary and sufficient conditions is suggested for existence of a dual bundle.