Accurate evaluation of layer potentials near boundaries and interfaces are needed in many applications, including fluid-structure interaction problems. A classical method to approximate the solution everywhere in the domain consists of using the same quadrature rule (Nyström method) used to solve the underlying boundary integral equation. This method is problematic for evaluations close to boundaries and interfaces. For a fixed number, N, of quadrature points, this method incurs a non-uniform error with O(1) errors in a boundary layer of thickness O(1/N). We have developed new asymptotic methods to remove this error. To demonstrate this method, we consider the Laplace problem and show the methods extended to the Stokes equations.