During the last few decades, integral equation methods have proven to be an indispensable tool for the robust and accurate solution of elliptic partial differential equations. For parabolic equations, there is a classical potential theory which leads to elegant integral equation methods used for analysis, but rarely for numerical computation because of the computational complexity associated with naive implementations. If numerical methods are to play a fundamental role in design and quantitative prediction, the solution techniques will have to be high-order accurate, robust, and fast. Moreover, especially when the governing equations involve nonlinear interactions, sharp features can develop in the solution even away from domain boundaries and automatic strategies will be required for refining or coarsening the grid in spacetime. 

In this talk, we present the frameworks of integral equation methods for both elliptic and parabolic equations, combining functional pieces including singular numerical quadratures, fast algorithms, and methods for automatic adaptivity. Potential applications and open problems for analysts and numerical analysts alike will also be mentioned.