Manifolds are a fundamental mathematical structure of central importance to geometry. The notion of cobordism has played an important role in their classification since Thom's work in the 1950s. In a different context, cobordisms are key to Atiyah and Segal's axiomatic approach to quantum field theory. We will explain how the two seemingly unrelated appearances of cobordisms have come together to give us a new approach to study the topology of manifolds and their diffeomorphisms. A key role has been played by the classifying space of the cobordism category. In turn in work of Freed-Hopkins, the classifying space of the cobordism category and certain maps from it have been interpreted as invertible field topological quantum field theory (TQFT). We will try to explain this circle of ideas.