  In this paper we study the general affine geometry of curves in affine space
$A^2$. For a regular plane curves we define two kinds of moving frames. The
first is of minimal order in all moving frames.The second is the Frenet moving
frame. We get the moving equations of these moving frames. And we prove that
curvature and signature are the complete invariants of regular curves. As
application we give a complete classification of constant curvature curves in
$A^2$.
