||In this talk we discuss the question whether rational homogeneous spaces are rigid under Fano deformation. In other words, given any smooth connected family f:X -> Z of Fano manifolds, if one fiber is biholomorphic to a rational homogeneous space S, whether is f an S-fibration? The cases of Picard number one were studied in a series of papers by J.-M. Hwang and N. Mok. For higher Picard number cases, we notice that the Picard number of a rational homogeneous space G/P is less or equal to the rank of G. Recently A. Weber and J. A. Wisniewski proved that rational homogeneous spaces G/P with Picard numbers equal to the rank of G (i.e. complete flag manifolds) are rigid under Fano deformation. We will study the Fano deformation rigidity of G/P whose Picard number equals to rank G-1.