  We deduce eigenvalue asymptotics of the Neumann--Poincar\'e operators in
three dimensions. The region $\Omega$ is $C^{2, \alpha}$ ($\alpha>0$) bounded
in ${\mathbf R}^3$ and the Neumann--Poincar\'e operator ${\mathcal
K}_{\partial\Omega} : L^2(\partial \Omega) \rightarrow L^2(\partial \Omega) $
is defined by $$ {\mathcal K}_{\partial\Omega}[\psi]({\bf x}) := \frac{1}{4\pi}
\int_{\partial \Omega} \frac{\langle {\bf y}-{\bf x}, {\bf n}({\bf y})
\rangle}{|{\bf x}-{\bf y}|^3} \psi({\bf y})\; dS_{\bf y} $$ where $dS_{\bf y}$
is the surface element and ${\bf n}({\bf y})$ is the outer normal vector on
$\partial \Omega$. Then the ordering eigenvalues of the Neumann--Poincar\'e
operator $\lambda_j ({\mathcal K}_{\partial \Omega})$ satisfy $$
|\lambda_j({\mathcal K}_{\partial \Omega})| \sim \Big\{\frac{3W(\partial
\Omega) - 2\pi \chi(\partial \Omega)}{128 \pi} \Big\}^{1/2} j^{-1/2}\quad
\text{as}\ j \rightarrow \infty. $$ Here $W(\partial \Omega)$ and
$\chi(\partial \Omega)$ denote, respectively, the Willmore energy and the Euler
charateristic of the boundary surface $\partial\Omega$. This formula is the
so-called Weyl's law for eigenvalue problems of Neumann--Poincar\'e operators.
