  We study the evaluation of layer potentials close to the domain boundary.
Accurate evaluation of layer potentials near boundaries is needed in many
applications, including fluid-structure interactions and near-field scattering
in nano-optics. When numerically evaluating layer potentials, it is natural to
use the same quadrature rule as the one used in the Nystr\"om method to solve
the underlying boundary integral equation. However, this method is problematic
for evaluation points close to boundaries. For a fixed number of quadrature
points, $N$, this method incurs $O(1)$ errors in a boundary layer of thickness
$O(1/N)$. Using an asymptotic expansion for the kernel of the layer potential,
we remove this $O(1)$ error. We demonstrate the effectiveness of this method
for interior and exterior problems for Laplace's equation in two dimensions.
