Blaschke introduced the term integral geometry to refer to an array  of interrelated identities among the integrals of certain natural  geometric quantities, the prototype being Crofton's formula for the  length of a plane curve as an integral over the space of straight  lines. In fact every compact group G acting transitively on a sphere gives rise to such an array.
Recent developments (Alesker's theory of convex valuations)  have revealed a beautiful algebraic structure underlying this array.  We will describe this theory in general, and sketch the structure in the particular cases of the orthogonal and unitary groups.

At the same time these ideas offer an approach to the study of  singular spaces, as the integrals involved tend to be rather  insensitive to the smoothness of the objects in question. The basic  construction is the normal cycle (also known as the characteristic cycle),  which is a Lagrangian integral current canonically associated to a singular subspace X. In many respects this association is still very poorly understood.