Let X ⊂ ℙn be a complex connected projective, non-degenerate, linearly normal manifold of degree d ≤ n. The main result of this paper is a classification of such manifolds. As a by-product of the classification it follows that these manifolds are either rational or Fano. In particular, they are simply connected (hence regular) and of negative Kodaira dimension. Moreover, easy examples show that d ≤ n is the best possible bound for such properties to hold true. The proof of our theorem makes essential use of the adjunction mapping and, in particular, the main result of  plays a crucial role in the argument.