Let \(\Omega_0\) be a bounded convex region in \(\mathbb{R}^{n+1}\) with smooth boundary \(M_0= \partial\Omega_0\). We denote by \(K\) the Gauss curvature of a hypersurface \(M\) in \(\mathbb{R}^{n+1}\) with the outward normal vector \(\nu\) for \(M\). The author studies the hypersurfaces moving by their Gauss curvature, motivated by several sources such as Gauss curvature flows, introduced by \textit{W. J. Firey} [Mathematica 21, 1-11 (1974; Zbl 0311.52003)], affine geometry, image analysis, gradient flows of the mean width as well as evolving hypersurfaces and degenerate fully nonlinear partial differential equations. The evolution equation studied by the author is of the following type: \(dx/dt= -\rho(\nu(x)) K(x)^\alpha \nu(x)\), where \(\rho\) is some positive function on \(S^n\). The author obtains the following theorem: For any smooth positive function \(\rho:S^n\to \mathbb{R}\) and any \(\alpha\in (1/(n+2),1/n]\) there exists a family of embeddings \(x: S^n\times [0,T)\to \mathbb{R}^{n+1}\) satisfying the above-differential equation such that \(M_t= x(S^n,t)\), converges in Hausdorff distance to \(M_0\) as \(t\to 0\). \(M_t\) is smooth and strictly convex for \(t>0\) and converges to a point \(p\in \mathbb{R}^{n+1}\) as \(t\to T\). Furthermore, the hypersurface \(\widetilde{M}_t= (\text{Vol} (S^n)/ \text{Vol} (M_t))^{1/(n+1)} (M_t-p)\) converges in \(C^\infty\) as \(t\to T\), to a smooth strictly convex limit hypersurface \(\widetilde{M}_T\) for which \(\langle x,\nu\rangle= c\cdot\rho(\nu) K^\alpha\) for some \(c>0\). The author also gives a short proof for the above theorem which says that \(M_t\) converges in \(C^\infty\) as \(t\to T\), to the ellipsoid centered at the origin when \(\alpha= 1/(n+2)\) and \(\rho\equiv 1\).