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 \(0\leq x_ 1+x_ 2+...+x_ p\leq \Delta\), where \(x_ 1,x_ 2,...,x_ p\) and \(\Delta\) are non-negative integers. Some further properties of the closely related multivariable continuous Hahn polynomials are also discussed.