This paper contains some results from the thesis written by the second author under the supervision of the first one. They study the so-called k-symmetric spaces, a class of Riemannian manifolds which generalizes properly the class of symmetric spaces. More precisely, the authors treat the following problem: which of the k-symmetric spaces can be realized as a k-symmetric submanifold in some \(E^ d ?\) (The notion of a ''realization'' has a specific meaning.) First they prove that each compact k-symmetric space has such a realization and they provide some nice examples. Then they derive some sufficient conditions for the non-compact case to be non-realizable. They derive several interesting conclusions from it and apply the theory for dimensions \(\leq 5\) to get an almost complete answer to the problem.