Summary: We obtain a complete classification of proper biharmonic hypersurfaces with at most three distinct principal curvatures in sphere spaces with arbitrary dimension. Precisely, together with known results of Balmuş-Montaldo-Oniciuc, we prove that compact orientable proper biharmonic hypersurfaces with at most three distinct principal curvatures in sphere spaces \(\mathbb{S}^{n+1}\) are either the hypersphere \(\mathbb{S}^n(1/\sqrt{2})\) or the Clifford hypersurface \(\mathbb{S}^{n_1}(1/\sqrt{2})\times\mathbb{S}^{n_2}(1/\sqrt{2})\) with \(n_1+n_2=n\) and \(n_1\neq n_2\). Moreover, we also show that there do not exist proper biharmonic hypersurface with at most three distinct principal curvatures in hyperbolic spaces \(\mathbb{H}^{n+1}\).