New physics has traditionally been expected in the high-pT region at high-energy collider experiments. If new particles are light and weakly coupled, however, this focus may be completely misguided: light particles are typically highly concentrated within a few mrad of the beam line, allowing sensitive searches with small detectors, and even extremely weakly coupled particles may be produced in large numbers there. We propose a new experiment, forward search experiment, or FASER, which would be placed downstream of the ATLAS or CMS interaction point (IP) in the very forward region and operated concurrently there. Two representative on-axis locations are studied: a far location, 400 m from the IP and just off the beam tunnel, and a near location, just 150 m from the IP and right behind the TAN neutral particle absorber. For each location, we examine leading neutrino- and beam-induced backgrounds. As a concrete example of light, weakly coupled particles, we consider dark photons produced through light meson decay and proton bremsstrahlung. We find that even a relatively small and inexpensive cylindrical detector, with a radius of ∼10 cm and length of 5-10 m, depending on the location, can discover dark photons in a large and unprobed region of parameter space with dark photon mass mA′∼10-500 MeV and kinetic mixing parameter ϵ∼10-6-10-3. FASER will clearly also be sensitive to many other forms of new physics. We conclude with a discussion of topics for further study that will be essential for understanding FASER's feasibility, optimizing its design, and realizing its discovery potential.