Summary: We deal with several characterizations of the Hardy-Sobolev spaces in the unit ball of \({\mathbb{C}}^ n\), that is, spaces of holomorphic functions in the ball whose derivatives up to a certain order belong to the classical Hardy spaces. Some of our characterizations are in terms of maximal functions, area functions or Littlewood-Paley functions involving only complex-tangential derivatives. A special case of our results is a characterization of \(H^ p\) itself involving only complex-tangential derivatives.