  We study various supersymmetric RG flows of $N=3$ Chern-Simons-Matter theory
in three dimensions by using four-dimensional $N=3$ gauged supergravity coupled
to eight vector multiplets with $SO(3)\times SU(3)$ gauge group. The $AdS_4$
critical point preserving the full $SO(3)\times SU(3)$ provides a gravity dual
of $N=3$ superconformal field theory with flavor symmetry $SU(3)$. We study the
scalar potential and identify a new supersymmetric $AdS_4$ critical point
preserving the full $N=3$ supersymmetry and unbroken $SO(3)\times U(1)$
symmetry. An analytic RG flow solution interpolating between $SO(3)\times
SU(3)$ and $SO(3)\times U(1)$ critical points is explicitly given. We then
investigate possible RG flows from these $AdS_4$ critical points to
non-conformal field theories in the IR. All of the singularities appearing in
the IR turn out to be physically acceptable. Furthermore, we look for
supersymmetric solutions of the form $AdS_2\times \Sigma_2$ with $\Sigma_2$
being a two-sphere or a two-dimensional hyperbolic space and find a number of
$AdS_2$ geometries preserving four supercharges with $SO(2)\times SO(2)\times
SO(2)$ and $SO(2)\times SO(2)$ symmetries.
