  For generalized KdV models with polynomial nonlinearity, we establish
nonlinear smoothing property in $H^s$ for $s>\frac{1}{2}$. Such smoothing
effect persists globally, provided that the $H^1$ norm does not blow up in
finite time. More specifically, we show that a translate of the nonlinear part
of the solution gains $\min(2s-1,1)-$ derivatives for $s>\frac{1}{2}$.
Following a new simple method, which is of independent interest, we establish
that, for $s>1$, $H^s$ norm of a solution grows at most by $\langle
t\rangle^{s-1+}$ if $H^1$ norm is a priori controlled.
