Gamma-ray bursts (GRBs) are the most powerful electromagnetic events in universe. GRBs are powered by either core-collapse of massive stars or binary mergers of two compact objects. These progenitor systems are believed to launch relativistic, collimated jets, which produce short, bright gamma-ray flashes (prompt emission) and long-lived, fading emission (afterglow) in the broad energy band from radio to gamma-rays. Even though the characteristics of the prompt emission and the afterglow have been vigorously studied, many details of the physics of GRBs remain uncertain. The Fermi Gamma-ray Space Telescope(Fermi) provides invaluable data for studying GRBs with the help of a very wide field of view and broad energy coverage from the hard X-ray to gamma-ray band. Fermi consists of two instruments, the Gamma-ray Burst Monitor (GBM; 8 keV–40 MeV) and the Large Area Telescope(LAT; 20 MeV– >300 GeV). In this thesis, I present dedicated analysis results on three bright GRBs: GRB 131108A, GRB 160709A, and GRB 190114C. Each of them shows its own evolution that includes the unusual and general features of GRBs. In addition, I performed two systematic studies using the full 10 year samples of LAT and GBM detected GRBs. For the first, I focused on the high-energy emission (>100 MeV) and its origin by tracking its temporal and spectral evolution. In the second, focusing on the prompt emission phase, I found an observational signature that originates in the geometry of the relativistic jet, which had been predicted but was previously unobserved.