Contrast to the hyperbolic mean curvature flows studied in [HKL], [LS], and [KW], a new hyperbolic curvature flow is proposed
for convex hypersurfaces. This flow is most suited when the Gauss curvature is involved. The equation satisfied by the graph
of the hypersurface under this flow gives rise to a new class of fully nonlinear Euclidean invariant hyperbolic equations.