We will discuss the evolution of complete non-compact hypersurfaces embedded in $\mathbb{R}$$^{n+1}$ by fully nonlinear extrinsic geometric flows. Special emphasis will be given to the Inverse Mean curvature flow (an example of ultra-fast diffusion) and to the Gauss curvature flow (an example of slow-diffusion). We will address the questions of the long time existence of the flow and the regularity of solutions. We will also discuss as to how the asymptotic behavior at spatial infinity of the initial data affects the time of existence.