The fundamental theorem of projective geometry says that a self-map of a projective space P(V) that sends lines to lines is induced by a semi-linear endomorphism of V.  The lines in the projective spaces over the complex numbers, the quaternions, and in the octonion plane turn out to be maximally curved spheres with respect to the rank one symmetric space metric. Nagano and Peterson asked what can be said about diffeomorphisms of symmetric spaces of compact type that preserve the class of maximally curved spheres. We will review some answers to this question and present an analogue of the fundamental theorem for generalized flag manifolds (R-spaces) of minimal type (joint work with Sergio Console).