  An alternating sign matrix, or ASM, is a $(0, \pm 1)$-matrix where the
nonzero entries in each row and column alternate in sign. We generalize this
notion to hypermatrices: an $n\times n\times n$ hypermatrix $A=[a_{ijk}]$ is an
{\em alternating sign hypermatrix}, or ASHM, if each of its planes, obtained by
fixing one of the three indices, is an ASM. Several results concerning ASHMs
are shown, such as finding the maximum number of nonzeros of an $n\times
n\times n$ ASHM, and properties related to Latin squares. Moreover, we
investigate completion problems, in which one asks if a subhypermatrix can be
completed (extended) into an ASHM. We show several theorems of this type.
