We introduce a new type of supermanifold modelled on the whole exterior algebra. The structure of the resulting objects is more simple than supermanifolds previously studied yet still reflects the typical problems in the subject. An analysis of the structure of these objects leads to a greater understanding of the structure of supermanifolds in general. In particular, we are able to give criteria for a supermanifold to admit a vector bundle over its core manifold as a covering manifold. We are also able to identify the vector bundle underlying the z-thickening of a C" manifold as a tensor power of the tangent bundle and to prove that compact supermanifolds admit no embedding.-, into their model spaces. The structure of those supermanifolds defined by the vanishing of G ** functions is also analyzed in depth. Finally, we are able to exhibit supermanifolds that do not admit vector bundles as covering manifolds and we point to the need to investigate the existence of compact simply connected supermanifolds.