  We demonstrate that the arbitrarily weak quenched disorder on the surface of
a system of continuous symmetry destroys long range order in the bulk, and,
instead, quasi-long range order emerges. Correlation functions are calculated
exactly for the two- and three-dimensional XY model with surface randomness via
the functional renormalization group. Even at strong quenched disorder the
three-dimensional XY model possesses topological order. We also determine
roughness of a domain wall in the presence of surface disorder.
