Elliptic or parabolic equations with discontinuous coefficients are seen in many disciplines such as electromagnetic, fluid dynamics and biophysics. For the accurate numerical solutions of these equations, it is important to enforce the known jumps in the solution and/or its gradient on the internal interfaces. Failing to do this would result in solutions of low accuracy and convergence of low order, or even divergence of the computation. In this talk I will briefly review the elliptic interface problems and their numerical algorithms. I will then introduce a novel matched interface and boundary (MIB) method and its applications to the solution of the electrostatic potential of macromolecules.