  We prove the sharp global well-posedness results for the initial value
problems (IVPs) associated to the modified Korteweg-de Vries (mKdV) equation
and a system modeled by the coupled modified Korteweg-de Vries equations
(mKdV-system). To obtain our results we use the second generation of the
modified energy and almost conserved quantities, more precisely, the refined
I-method introduced by Colliander, Keel, Staffilani, Takaoka and Tao.
