A symmetric $R$-space is a compact Riemannian symmetric space which is realized as a linear isotropy orbit of certain Riemannian symmetric space of compact type. Since every Hermitian symmetric space of compact type is realized as an adjoint orbit of a compact semisimple Lie group, it is a symmetric $R$-space. In 1965 T. Nagano introduced the notion of symmetric $R$-space as a compact Riemannian symmetric space endowed with transitive action of noncompact Lie group which contains the isometry group of $M$. Then, S. Kobayashi constructed explicitly the embedding of some symmetric $R$-spaces into the Euclidean space. In his joint paper with M. Takeuchi in 1968, they proved that every $R$-space, which is a kind of generalization of a symmetric $R$-space, has a natural embedding into the Euclidean space which is a minimum embedding. Since then, a lot of researches on symmetric $R$-spaces as submanifolds in the Euclidean space have been done up to now. In my recent joint work with H. Tasaki we investigated fundamental properties of antipodal sets in a symmetric $R$-space by making use of the natural embedding. An antipodal set in a compact symmetric space is a finite subset in which the geodesic symmetry at each point is the identity. We also proved that the intersection of two real forms in a Hermitian symmetric space of compact type is an antipodal set, where it is known that every real form in a Hermitian symmetric space of compact type is a symmetric $R$-space, and vice versa.