The authors consider a generic vector field \(X\) on a smooth manifold \(M\) near a generically embedded submanifold \(Q\). They obtain a classification of such vector fields by giving local normal forms near a point in the image of \(Q\). The classification is obtained by means of the following ideas: Since \(X\) is a generic vector field, \(X\) has only isolated singularities and is therefore equal to \({\partial \over \partial x^ 1}\) in some local coordinates near a point in the image of a generic embedding of \(Q\). The image of \(Q\) is locally given as the zero set of a local submersion. Using singularity theory, the authors then classify such submersions under the group of local diffeomorphisms preserving \({\partial \over \partial x^ 1}\). From the normal forms obtained for such submersions they find local coordinates for the image of \(Q\) and for \(M\), transforming \({\partial \over \partial x^ 1}\) into the normal forms of the classification. Finally, using the same ideas, the authors give a similar classification for generic vector fields near a point in a generic hypersurface with boundary, or in a piecewise-smooth hypersurface.