We roughly investigate deformation rigidity problem of homogeneous varieties with Picard number one, VMRT theory which is related, and the rigidity results by Hwang, Mok and others. After, we introduce horospherical varieties with Picard number one and symmetric varieties with Picard number one which are spherical varieties. These are quasi-homogeneous and finally, we study deformation rigidity of the above quasi-homogeneous varieties with Picard number one except the Cayley Grassmannian.