We examine edge decompositions of complete uniform hypergraphs whose parts are permuted transitively by a permutation of the vertex set. Such decompositions are called cyclic. We determine the cycle type of the permutations which yield cyclic decompositions, and consequently determine the feasible orders of complete $k$-uniform hypergraphs which admit cyclic decompositions with $t$ isomorphic parts. This leads to an algorithm for generating all such decompositions of a fixed feasible order. We then consider using the orbits of a given permutation on an $n$ element set on the $k$-subsets of the set to construct more general hypergraph decompositions, which may have non-isomorphic parts, and we relate the cycle type of the permutation to the number of isomorphism classes in the decompositions we obtain. In addition, we present an algebraic method for constructing cyclic hypergraph decompositions on finite fields which is related to the well known Paley graph construction. The construction is derived

from a partition of the cosets of a group and a function from the power set of the vertex set into this group. Several examples will be constructed using different groups and we discuss the symmetry and other properties of the hypergraphs we obtain.