  According to the AdS4/CFT3 correspondence, N=2 supersymmetric Chern-Simons
matter theories should have a stable fixed point in the infrared. In order to
support this prediction we study RG flows of two-level Chern-Simons matter
theories with/without flavors induced by the most general marginal
superpotential compatible with N=2 supersymmetry. At two loops we determine the
complete spectrum of fixed points and study their IR stability. Our analysis
covers a large class of models including perturbations of the ABJM/ABJ theories
with and without flavors, N=2,3 theories with different CS levels corresponding
to turning on a Romans mass and beta-deformations. In all cases we find curves
(or surfaces) of fixed points which are globally IR stable but locally unstable
in the following sense: The system has only one direction of stability which in
the ABJM case coincides with the maximal global symmetry preserving
perturbation, whereas along any other direction it flows to a different fixed
point on the surface. The question of conformal invariance vs. finiteness is
also addressed: While in general vanishing beta-functions imply two-loop
finiteness, we find a particular set of flavored theories where this is no
longer true.
