The main problem treated in this book is the following basic question: is it possible to characterize metrics on a given symmetric space of a compact type by means of the spectrum of its Laplacian? The book is devoted to a study of the infinitesimal version of the problem. In particular, infinitesimal isospectral deformations belong to the kernel of a certain Radon transform (defined in terms of integration over the flat totally geodesic tori of dimension equal to the rank of the space). A special version of the problem is to determine all symmetric spaces of compact type for which the Radon transform is injective in a suitable sense. The Guillemin criterion for infinitesimal spectral rigidity is also described and new methods for studying this type of rigidity are introduced. Projective spaces and real, complex and quaternionic Grassmanians are presented as main examples. There is a discussion and proofs of rigidity of Grassmanians. In the last part non-rigidity of the product of irreducible symmetric spaces is proved. The methods used in the book include harmonic analysis on homogeneous spaces and a resolution of the sheaf of Killing vector fields constructed by means of the theory of linear overdetermined PDEs.