FASER, the ForwArd Search ExpeRiment, is a proposed experiment dedicated to searching for light, extremely weakly-interacting particles at the LHC. Such particles may be produced in the LHC's high-energy collisions in large numbers in the far-forward region and then travel long distances through concrete and rock without interacting. They may then decay to visible particles in FASER, which is placed 480 m downstream of the ATLAS interaction point. In this work, we describe the FASER program. In its first stage, FASER is an extremely compact and inexpensive detector, sensitive to decays in a cylindrical region of radius R = 10 cm and length L = 1.5 m. FASER is planned to be constructed and installed in Long Shutdown 2 and will collect data during Run 3 of the 14 TeV LHC from 2021-23. If FASER is successful, FASER 2, a much larger successor with roughly R ~ 1 m and L ~ 5 m, could be constructed in Long Shutdown 3 and collect data during the HL-LHC era from 2026-35. FASER and FASER 2 have the potential to discover dark photons, dark Higgs bosons, heavy neutral leptons, axion-like particles, and many other long-lived particles, as well as provide new information about neutrinos, with potentially far-ranging implications for particle physics and cosmology. We describe the current status, anticipated challenges, and discovery prospects of the FASER program.