The Neptune Odyssey mission concept is a Flagship-class orbiter and atmospheric probe to the Neptune-Triton system. This bold mission of exploration would orbit an ice-giant planet to study the planet, its rings, small satellites, space environment, and the planet-sized moon Triton. Triton is a captured dwarf planet from the Kuiper Belt, twin of Pluto, and likely ocean world. Odyssey addresses Neptune system-level science, with equal priorities placed on Neptune, its rings, moons, space environment, and Triton. Between Uranus and Neptune, the latter is unique in providing simultaneous access to both an ice giant and a Kuiper Belt dwarf planet. The spacecraft - in a class equivalent to the NASA/ESA/ASI Cassini spacecraft - would launch by 2031 on a Space Launch System or equivalent launch vehicle and utilize a Jupiter gravity assist for a 12 yr cruise to Neptune and a 4 yr prime orbital mission; alternatively a launch after 2031 would have a 16 yr direct-to-Neptune cruise phase. Our solution provides annual launch opportunities and allows for an easy upgrade to the shorter (12 yr) cruise. Odyssey would orbit Neptune retrograde (prograde with respect to Triton), using the moon's gravity to shape the orbital tour and allow coverage of Triton, Neptune, and the space environment. The atmospheric entry probe would descend in ~37 minutes to the 10 bar pressure level in Neptune's atmosphere just before Odyssey's orbit-insertion engine burn. Odyssey's mission would end by conducting a Cassini-like "Grand Finale,"passing inside the rings and ultimately taking a final great plunge into Neptune's atmosphere.