Abstract: Ricci solitons are self similar solutions to the Ricci flow and a natural generalization of Einstein metrics. In this talk I'll survey some prior results about the classification of gradient shrinking Ricci solitons under varying conditions on dimension, curvature, or symmetry. Then I'll discuss a new classification result of 4-dimensional gradient shrinking Ricci solitons with half harmonic Weyl curvature. This is joint work with J.Y. Wu of Shanghai Maritime University and P. Wu of Cornell.