We establish short-time existence of solutions to the 3D Euler equations in uniformly local Sobolev spaces. The main challenge in this setting is the invalidity of the Biot-Savart Law due to the lack of spatial decay of vorticity at infinity. We derive a replacement identity from the Biot-Savart Law which is valid for functions in our uniformly local spaces. We also derive a pressure identity for our solutions. Finally, we obtain a blow-up criterion which implies global existence of our solutions in 2D and which is analogous to the Beale-Kato-Majda blow-up criterion for the Euler equations in the classical Sobolev spaces. This is joint work with David Ambrose, Daniel Erickson, and Jim Kelliher.