In this paper we prove: 1) In characteristic p > 0 every simplicial toric affine or projective variety with full parametrization is a set-theoretic complete intersection. This extends previous results by R. Hartshorne (1979) and T. T. Moh (1985). 2) In any characteristic, every simplicial toric affine or projective variety with full parametrization is an almost set-theoretic complete intersection. This extends previous known results by M. Barile and M. Morales (1998) and A. Thoma (to appear). 3) In any characteristic, every simplicial toric affine or projective variety of codimension two is an almost set-theoretic complete intersection. Moreover the proofs are constructive and the equations we find are binomial ones.