Summary: A compact quantum hypergroup is a unital \(C^*\)-algebra equipped with a completely positive coassociative coproduct. The most important examples of such a structure are associated with double cosets of compact matrix pseudogroups in the sense of S. L. Woronowicz. We give a precise definition of a compact quantum hypergroup; prove existence and uniqueness of the Haar measure, establish orthogonality relations for matrix elements of irreducible corepresentations and construct a Peter-Weyl theory for irreducible corepresentations.