One consequence of the celebrated theorem of Beurling is that all non-zero submodules of H^2(D) are unitarily equivalent to H^2(D). For submodules of H^2(D^n) over A(D^n), n>1, some are unitarily equivalent to H^2(D^n) and some are not. For the Hardy module H^2(partial(B)^n) over the ball algebra A(B^n), the existence of inner functions on B^n established the existence of proper submodules of H^2(partial(B)^n) that are unitarily equivalent to H^2(B^n). For the Bergman modules over the polydisk or the ball, one can show that no proper submodule is unitarily equivalent to the Bergman module itself. In this talk we consider the question of which Hilbert modules have proper submodules unitarily equivalent to the original.