  We prove that the Korteweg-de Vries initial-value problem is globally
well-posed in $H^{-3/4}(\R)$ and the modified Korteweg-de Vries initial-value
problem is globally well-posed in $H^{1/4}(\R)$. The new ingredient is that we
use directly the contraction principle to prove local well-posedness for KdV
equation at $s=-3/4$ by constructing some special resolution spaces in order to
avoid some 'logarithmic divergence' from the high-high interactions. Our local
solution has almost the same properties as those for $H^s (s>-3/4)$ solution
which enable us to apply the I-method to extend it to a global solution.
