Manifolds are objects that locally look like Euclidean space, but may be something else. This notion carries over to number theory in a variety of ways, from finding trivial proofs that certain Diophantine equations have no solutions to far more sophisticated settings such as the recent work of Bhargava et al. on the conjecture of Birch and Swinnerton-Dyer, the older work of Wiles and Taylor-Wiles on the Shimura-Taniyama Conjecture and even the *much* older work of Kummer.