When a massive star explodes as a supernova, it injects a huge amount of energy into its surroundings. The resultant expanding blast-wave and its interaction with the surrounding medium is known as a supernova remnant (SNR). The shock created by the supernova event is believed to be the primary accelerator of cosmic rays (CRs) in our Galaxy. While SNRs are observable across the electromagnetic spectrum, studying the gamma-ray emission from these sources is crucial in understanding the origin of CRs and acceleration processes acting therein. Recent advances in gamma-ray astronomy present new opportunities to study the aftermath of stellar explosions at gamma-ray energies. In 2008 the Fermi Gamma-Ray Space Telescope was launched into orbit and, with its unmatched gamma-ray resolution, has opened up a new window on the high-energy sky. In this thesis, I present new work using data from the primary instrument on the Fermi observatory, the Large Area Telescope (LAT), to study both individual SNRs as well as the population of remnants observable by the LAT, with a focus on searching for spatially extended emission from these remnants. To uniformly determine the high-energy properties of SNRs, I developed an automated method to systematically characterize the gamma-ray emission in a region of the sky. Applying this method to the locations of several hundred radio-observed SNRs, we classified 30 gamma-ray sources as likely being associated with SNRs. Our results, combined with archival radio, X-ray, and TeV observations, serve to challenge previously sufficient, simple gamma-ray SNR emission models. I also present a study of the sources detected above 50 GeV, focusing on those lying in the Galactic plane. 31 sources were shown to be significantly spatially extended with 5 of those being newly detected. Finally, I present a dedicated analysis of one of the 5 newly detected extended sources. I determined that the extended GeV emission likely originated from the shock of SNR G150.3+4.5. Combined with archival radio and X-ray data, I consider several possible origin scenarios, including one in which the SNR may be one of the youngest, closest gamma-ray SNRs detected by the LAT.