A continuous function f(x,y) is given on the unit square, and it is desired to approximate it in the Tchebycheff sense by a function of the form g(x) + h(y). Several aspects of this problem are studied here. New results are obtained for the Dili-berto-Straus Algorithm, and some examples show how rapidly or how slowly it converges. New estimates are derived for the “degree” of approximation. Further results concern: (1) the description of the set of all best approximations; (2) the special case when f is differentiable; and (3) the least-squares version of the same problem.