Abstract Prof. Dawei Chen, University of Illinois at Chicago, Nov. 4, 2008, 4 pm, TTC 245 A twisted cubic is a degree three smooth rational curve in P³. There are several different compactifications for the space of twisted cubics, such as the Kontsevich space of stable maps, the Hilbert scheme, the Chow variety, etc. We study the geometry of those compactifications and display their inter-connections from the viewpoint of Mori's theory.