There is a symmetry at each point in a Riemannian symmetric space. An antipodal set is a subset where the restriction of the symmetry at each point is the identity, which was introduced by Chen and Nagano. A set of two antipodal points (in a usual sense) in a sphere is a typical example of antipodal sets. A maximal antipodal set is a kind of frame of a compact Riemannian symmetric space. In this talk I mainly explain antipodal sets in symmetric R-spaces and oriented real Grassmann manifolds.