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

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... , 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.