The class of quantum hypergraph states is introduced and discussed. We show that these n-qubit states can be regarded as a generalization of well-known graph states since they are defined according to an underlying mathematical hypergraph, i.e. a graph where edges connecting more than two vertices are considered. We develop a generalized stabilizer formalism in order to describe them and discuss inequivalence under local unitaries. We then prove that this class of states describes exhaustively the states employed in quantum algorithms, such as Deutsch-Jozsa’s and Grover’s. Finally we discuss some possible further developments, especially in connection with one-way quantum computing.