  We develop a theory of parametrized geometric cobordism by introducing smooth
Thom stacks. This requires identifying and constructing a smooth representative
of the Thom functor acting on vector bundles equipped with extra geometric
data, leading to a geometric refinement of the the Pontrjagin-Thom construction
in stacks. We demonstrate that the resulting theory generalizes the
parametrized cobordism of Galatius-Madsen-Tillman-Weiss. The theory has the
feature of being both versatile and general, allowing for the inclusion of
families of various geometric data, such as metrics on manifolds and
connections on vector bundles, as in recent work of Cohen-Galatius-Kitchloo and
Ayala.
