In this talk, we will discuss 4-dimensional complete (not necessarily compact) gradient shrinking Ricci solitons. We will prove that a 4-dimensional complete gradient shrinking Ricci soliton satisfying a pointwise condition involving either the self-dual (or anti-self-dual) part of the Weyl tensor is either Einstein, or a finite quotient of either the Gaussian shrinking soliton R^4, or S^3×R, or S^2×R^2. Moreover, we show that if the quotient of norm of the self-dual part of the Weyl tensor and scalar curvature is close to that on a Kähler metric in a specific sense, then the gradient Ricci soliton must be either half- conformally flat or locally Kähler. This is a joint work with Xiaodong Cao and Hung Tran.