We provide characterizations for boundedness of multilinear Fourier multiplier operators on Hardy or Lebesgue spaces with symbols locally in Sobolev spaces. Let H-q (R-n) denote the Hardy space when 0 < q <= 1 and the Lebesgue space L-q(R-n) when 1 < q <= infinity. We find optimal conditions on m-linear Fourier multiplier operators to be bounded from H-P1 x...x H-Pm to L-P when 1/p = 1/p(1) +...+ 1/p(m) in terms of local L-2-Sobolev space estimates for the symbol of the operator. Our conditions provide multilinear analogues of the linear results of Calderon and Torchinsky [1] and of the bilinear results of Miyachi and Tomita [17]. The extension to general m is significantly more complicated both technically and combinatorially; the optimal Sobolev space smoothness required of the symbol depends on the Hardy-Lebesgue exponents and is constant on various convex simplices formed by configurations of m2(m-1) 1 points in [0, infinity)(m).