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