  In this paper, we investigate the H\"ormander type theorems for the
multi-linear and multi-parameter Fourier multipliers. When the multipliers are
characterized by $L^u$-based Sobolev norms for $1<u\le 2$ , our results on the
smoothness assumptions are sharp in the multi-parameter and bilinear case. In
the multi-parameter and multi-linear case, our results are almost sharp.
Moreover, even in the one-parameter and multi-linear case, our results improve
earlier ones in the literature.
