As well as arising naturally in the study of non-intersecting random paths, random spanning trees, and eigenvalues of random matrices, determinantal point processes (sometimes also called fermionic point processes) are relatively easy to simulate and provide a quite broad class of models that exhibit repulsion between points. The fundamental ingredient used to construct a determinantal point process is a kernel giving the pairwise interactions between points: the joint distribution of any number of points then has a simple expression in terms of determinants of certain matrices defined from this kernel. In this paper we initiate the study of an analogous class of point processes that are defined in terms of a kernel giving the interaction between 2M points for some integer M . The role of matrices is now played by 2M -dimensional hypercubic arrays, and the determinant is replaced by a suitable generalization of it to such arrays Cayleys first hyperdeterminant. We show that some of the desirable features of determinantal point processes continue to be exhibited by this generalization.