  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
Balmu\c{s}-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 $\mathbb S^n(1/\sqrt2)$ or the
Clifford hypersurface $\mathbb S^{n_1}(1/\sqrt2)\times\mathbb
S^{n_2}(1/\sqrt2)$ with $n_1+n_2=n$ and $n_1\neq n_2$. Moreover, we also show
that there does not exist proper biharmonic hypersurface with at most three
distinct principal curvatures in hyperbolic spaces $\mathbb H^{n+1}$.
