A variety of applications in electromagnetic scattering, circuit simulations, and vorticity methods for fluid flows require solution of dense linear systems of equations. The large storage and computational requirements of these systems limits the size of problems that can be solved. Recent work on approximation techniques for particle dynamics coupled with advances in parallel computing have enabled solution of problems several orders of magnitude larger. Using methods such as Fast Multipole and Barnes-Hut, we have been able to solve dense linear systems with hundreds of thousands of variables in minutes on massively parallel computers.