  Given solenoidal vector $u_0\in H^{-2\d}\cap H^1(\R^2),$ $\r_0-1\in
L^2(\R^2),$ and $\r_0 \in L^\infty\cap\dot{W}^{1,r}(\R^2)$ with a positive
lower bound for $\d\in (0,\f12)$ and $2<r<\f{2}{1-2\d},$ we prove that 2-D
incompressible inhomogeneous Navier-Stokes system \eqref{1.1} has a unique
global solution provided that the viscous coefficient $\mu(\r_0)$ is close
enough to 1 in the $L^\infty$ norm compared to the size of $\d$ and the norms
of the initial data. With smoother initial data, we can prove the propagation
of regularities for such solutions. Furthermore, for $1<p<4,$ if $(\r_0-1,u_0)$
belongs to the critical Besov spaces $\dB^{\f2p}_{p,1}(\R^2)\times
\bigl(\dB^{-1+\f2p}_{p,1}\cap L^2(\R^2)\bigr)$ and the $\dB^{\f2p}_{p,1}(\R^2)$
norm of $\r_0-1$ is sufficiently small compared to the exponential of
$\|u_0\|_{L^2}^2+\|u_0\|_{\dB^{-1+\f2p}_{p,1}},$ we prove the global
well-posedness of \eqref{1.1} in the scaling invariant spaces. Finally for
initial data in the almost critical Besov spaces, we prove the global
well-posedness of \eqref{1.1} under the assumption that the $L^\infty$ norm of
$\r_0-1$ is sufficiently small.
