The Calculus of Moving Surfaces (CMS) is an extension of classical differential geometry to moving manifolds. It follows the logical framework of Ricci's and Levi-Civita's absolute differential calculus. Central to the CMS is the δ/δt-derivative that plays a role analogous to that of the covariant derivative ∇_{α} on the differential manifold. In particular, it has the property that it produces a tensor when applied to a tensor.

The CMS is a powerful technique for analyzing problems with moving boundaries. I will give an overview of the CMS and illustrate its capability by demonstrating a number of applications. Applications will include:

1. Boundary variation problems -- what is the change in solution of a boundary value problem induced by a change in shape?

2. Shape optimization problems -- what shape delivers an extremal value of a shape dependent objective function?

3. Dynamic problems -- I am excited to present the recently proposed exact nonlinear equations of fluid film dynamics. Derived from the Least Action Principle, these equations are a direct analogue of the Navier-Stokes equations and therefore possess the same key characteristics: conservation of mass and, in the case of inviscid equations, conservation of energy, pointwise conservation of vorticity and conservation of circulation around a closed loop.