Generalized Chen submanifolds or \(k\)-th Chen submanifolds are defined. The authors give a characterization of those submanifolds in terms of an operator of J. Simons. They relate these submanifolds to submanifolds of finite type introduced by B. Y. Chen and prove that: Let \(M\) be a compact submanifold in \(E^ m\) with parallel second fundamental form; then \(M\) is of finite \(k\)-type if and only if \(M\) is a \(k\)-th Chen submanifold in \(E^ m\).