We prove that several definitions of integral multilinear mappings given in the literature are equivalent. We show that what we call S-factorizable multilinear mappings are integral, but that the converse is not true (contrary to earlier claims). Using S-factorizable polynomials we give characterizations of L∞-spaces, Asplund spaces, spaces not containing ℓ1 and spaces with the compact range property. Some of these characterizations seem to be new even for linear operators.