In this talk, we review classical results on flows by powers of the Gauss curvature in \$\mathbb{R}^{n+1}\$. For large powers, the equation exhibits more degeneracy, while for small powers it becomes more singular. The critical exponent is 1/(n+2), where convergence to ellipsoids was established by Andrews. For all supercritical exponent, convergence to the sphere was proved by Andrews–Guan–Ni and Brendle–Choi–Daskalopoulos. We then discuss the more challenging problem of evolving hypersurfaces in general ambient spaces, where the curvature of the ambient manifold will interfere with the motion. Huisken introduced a suitable “convex enough” condition, depending on the ambient geometry, for mean curvature flow in general Riemannian manifolds. For flows by supercritical powers of the Gauss curvature in space forms, we show that this additional “convex enough” assumption can be removed, and we prove convergence under the same condition imposed in \$\mathbb{R}^{n+1}\$, namely strict convexity of the initial hypersurface. The key is to establish an almost monotonicity formula and an almost invariance property.