<p>We introduce a new type of null hypersurfaces in a Lorentzian manifold, which are called weakly parallel null hypersurfaces. It resembles the notion of parallel (nondegenerate) hypersurface and it includes the totally geodesic and totally umbilical ones. Like the latter, the weakly parallel condition is an invariant property of the null hypersurface (it does not depend on any choice). The weakly parallel condition is related to the existence of a vector field in a Riemannian manifold with certain property, which is called exponential vector field. This relationship is used to show some properties of weakly parallel null hypersurfaces. For example, the classification of the exponential vector fields in the Euclidean space leads to a classification of the weakly parallel null hypersurfaces in the Minkowski space.</p>