  A Riemannian cone $(C, g_C)$ is by definition a warped product $C =
\mathbb{R}^+ \times L$ with metric $g_C = dr^2 \oplus r^2 g_L$, where $(L,g_L)$
is a compact Riemannian manifold without boundary. We say that $C$ is a
Calabi-Yau cone if $g_C$ is a Ricci-flat K\"ahler metric and if $C$ admits a
$g_C$-parallel holomorphic volume form; this is equivalent to the cross-section
$(L,g_L)$ being a Sasaki-Einstein manifold. In this paper, we give a complete
classification of all smooth complete Calabi-Yau manifolds asymptotic to some
given Calabi-Yau cone at a polynomial rate at infinity. As a special case, this
includes a proof of Kronheimer's classification of ALE hyper-K\"ahler
$4$-manifolds without twistor theory.
