  We define parametrized cobordism categories and study their formal properties
as bivariant theories. Bivariant transformations to a strongly excisive
bivariant theory give rise to characteristic classes of smooth bundles with
strong additivity properties. In the case of cobordisms between manifolds with
boundary, we prove that such a bivariant transformation is uniquely determined
by its value at the universal disk bundle. This description of bivariant
transformations yields a short proof of the Dwyer-Weiss-Williams family index
theorem for the parametrized A-theory Euler characteristic of a smooth bundle.
