The Gauss curvature flow is an evolution of convex hypersurface by Gaussian curvature and it models the wearing process undergone by a pebble. The classification of solitons and asymptotic behavior of flow have been important questions for decades. 

After introducing previous results on compact flows, the goal is to present our recent result on the asymptotic behavior of non-compact flow: if complete convex non-compact initial hypersurface is defined in a cylinder of bounded cross-section, the flow converges to a translating soliton which is uniquely determined by the cylinder asymptotic to the initial hypersurface. This is a joint work with Kyeongsu Choi and Panagiota Daskalopoulos.