The main topic treated in this book is a further study of (infinitesimal) spectral deformations of Riemannian symmetric spaces of compact type. Both of the authors have treated this topic and its connection to the Radon transform already in other papers and in their previous monograph (published in the Princeton Ann. of Math. Studies). Here they concentrate on the case of real Grassmannian manifolds Gn,2n of n dimensional planes in R2n and its reduced version Ğn,2n (the quotient of Gn,2n by the action sending an n-plane to its orthogonal complement). It is shown in the book that for n=3 the Grassmannian Ğ3,3 admits nontrivial isospectral deformations. Another question treated in the book is a study of properties of invariant differential forms on Ğn,2n. The authors study necessary and sufficient conditions for exactness of forms in terms of a Radon transform.