The Wasserstein space of a compact Riemannian manifold is the set of Borel probablity measures endowed with a distance defined using optimal transportation (roughly, the distance between two measures is the least cost needed to transport one to the other, given that it costs $m d^2$ to move an amount $m$ of mass by a distance $d$.) It has been noticed by Otto that this Wasserstein space can be considered an infinite-dimensional manifold. He used this insight to convert some PDEs into gradient flows on the Wasserstein space, the goal being to get existence and uniqueness result more easily. I should add that Gigli recently proposed a rigorous framework for the differentiable structure of Wasserstein spaces.