The notion of symmetry underlies a large number of new ideas and major advances in Science,Engineering and Art. From the mathematical viewpoint, the intuitive idea of symmetry as thebalanced correspondence of shape along space translates to the existence of a transformationgroup acting on such space. The rst natural eld for the study of symmetry is then geometry.Conversely, in his inuential Erlanger Programm, Felix Klein described geometry as the study ofthose properties of a space that are invariant under a transformation group. Hence, symmetry liesin the very core of geometry.In Riemannian geometry, the natural group to consider is the isometry group, that is, thegroup of those transformations of the space that preserve distances. The action of a subgroupof the isometry group of a given manifold is called an isometric action. Its cohomogeneity is thelowest codimension of its orbits. Each one of the orbits of such an isometric action is called an(extrinsically) homogeneous submanifold, and the collection of all the orbits is the orbit foliationof the action.The main objects of study in this thesis are certain kinds of submanifolds with a particularlyhigh degree of symmetry. Our ultimate goal is to decide whether the intuitive notion of symmetryis reected in the mathematical notion of symmetry, namely if the correspondence of shape atdi erent parts of the submanifold implies that the submanifold is homogeneous.