  We discuss the renormalization group improved effective action and running
surface couplings in curved spacetime with boundary. Using scalar
self-interacting theory as an example, we study the influence of the boundary
effects to effective equations of motion in spherical cap and the relevance of
surface running couplings to quantum cosmology and symmetry breaking
phenomenon. Running surface couplings in the asymptotically free SU(2) gauge
theory are found.
