Symmetric spaces appear in various branches of mathematics and physics. Their origins go back to Cartan’s influential work on Riemannian geometry. In this talk after briefly reviewing symmetric spaces and their origins, we will discuss a special family of Hessenberg varieties in relation with K-orbit closures in flag varieties, where K is the symmetric subgroup S(GL(k)xGL(n-k)) in SL(n). Our goal is to explain how K-orbits can be used for understanding the geometry of Hessenberg varieties of semisimple operators with two eigenvalues. This is a joint work with John Shareshian, Martha Precup, and Ozlem Ugurlu