A multivariable biorthogonal generalization of the discrete Hahn polynomials, a p+1 complex parameter family, where p is the number of variables, is presented. It is shown that the polynomials are orthogonal with respect to subspaces of lower degree and biorthogonal within a given subspace. These properties are over the discrete simplex 0≤x1+x2+⋅⋅⋅+xp≤Δ, where x1, x2,...,xp and Δ are non-negative integers. Some further properties of the closely related multivariable continuous Hahn polynomials are also discussed.