In this talk, I will discuss on lattice formulations of 2D N=(2,2) SQCD
preservng one of the supercharges.
In particular, the overlap Dirac operator, which satisfies the Ginsparg-Wilson
relation, is introduced to the matter sector of the theory.
It realizes the exact chiral flavor symmetry on the lattice, to make possible
to define the lattice action for general number of the flavors of fundamental
and anti-fundamental matter multiplets and for general twisted masses.
Furthermore, superpotential terms can be introduced with exact holomorphic
or anti-holomorphic structure on the lattice.
I will also discuss the lattice formulation of matter multiplets charged
only under the central U(1) (the overall U(1)) of the gauge group G=U(N),
and then construct lattice models for gauged linear sigma models with
exactly preserving one supercharge and their chiral flavor symmetry.