  We study higher-rank Radon transforms that take functions on $j$-dimensional
totally geodesic submanifolds in the $n$-dimensional real constant curvature
space to functions on similar submanifolds of dimension $k >j$. The
corresponding dual transforms are also considered. The transforms are explored
the Euclidean case (affine Grassmannian bundles), the elliptic case (compact
Grassmannians), and the hyperbolic case (the hyperboloid model, the
Beltrami-Klein model, and the projective model). The main objectives are sharp
conditions for the existence and injectivity of the Radon transforms in
Lebesgue spaces, transition from one model to another, support theorems, and
inversion formulas. Conjectures and open problems are discussed.
