Alexandrov-Fenchel inequality can be viewed as a higher order generalization of the isoperimetric inequality. It reveals the relation between the quermassintegrals. In this talk, we review the concept of curvature integrals and quermassintegrals for closed hypersurfaces in the hyperbolic space, and compare them with that in the Euclidean space. Using the longtime existence and convergence result of a preserving quermassintegral curvature flow, we derive a large class of optimal Alexandrov-Fenchel inequalities for horospherical convex hypersurfaces in the hyperbolic space. This is joint work with Prof. Guofang Wang.