Since its introduction in a series of papers published in 2010, the Fourier Continuation (FC) method for treating partial differential equations has shown great promise. The method, which utilizes an efficient one-dimensional periodic extension to enable the use of Fourier space methods in non-periodic settings while avoiding the Gibbs phenomenon, is especially well-suited to long-range wave propagation and transport problems. This talk will present the basic constructions underlying FC solvers and summarize some of the most interesting properties of these solvers, including exceptionally accurate dispersion curves, efficient parallelization, and optimal CFL scaling. Additionally, several example applications to which FC solvers have been successfully applied, including models for ultrasound cancer therapies, scattering of chirped radar signals, and complex fluid flows, will be presented.