Curvature-adapted submanifolds of symmetric spaces Thomas Murphy 53C1253C40curvature-adapted submanifoldsisoparametric hypersurfacesCayley projective and hyperbolic planescomplex two-plane Grassmannians Curvature-adapted submanifolds have been extensively studied in complex and quaternionic space forms. This paper extends their study to a wider class of ambient spaces. We generalize Cartan's theorem classifying isoparametric hypersurfaces of spheres to any compact symmetric space. Our second objective is to investigate such hypersurfaces in some specific symmetric spaces. We classify those with constant principal curvatures in the octonionic planes. Various classification results for hypersurfaces in complex two-plane Grassmannians are also obtained. Indiana University Mathematics Journal 2012 text pdf 10.1512/iumj.2012.61.4859 10.1512/iumj.2012.61.4859 en Indiana Univ. Math. J. 61 (2012) 831 - 847 state-of-the-art mathematics http://iumj.org/access/