For a $/mathbb{Q}$-Fano $3$-fold $X$ on which $K_X$ is a canonical divisor, we investigate the geometry inspired by the linear system $|-mK_X|$ in this paper and prove that the anti-$m$-canonical map  $/varphi_{-m}$ is birational onto its image for all $m/geq 39$.  By a weak $/mathbb{Q}$-Fano 3-fold $X$ we mean a projective one with at worst terminal singularities on which $-K_X$ is $/bQ$-Cartier, nef and big. For weak $/mathbb{Q}$-Fano 3-folds, we prove that  $/varphi_{-m}$ is birational onto its image for all $m/geq 97$.