We begin by recalling the invariant measures on the linear and affine Grassmannians \(G(d,k)\) and \(A(d,k)\) . We then derive and discuss Blaschke–Petkantschin-type transformation formulas. The affine version of this formula allows to rewrite an integral over tuples of points in \(\mathbb {R}^d\) as a double integral in which we first integrate over all affine subspaces A and then over all tuples of points spanning A, with an explicit Jacobian weight. The integral-geometric tools introduced in this chapter will be indispensable in the probabilistic and geometric constructions of the rest of the book.