Terahertz radiation has uses in applications ranging from security to medicine(1). However, sensitive room-temperature detection of terahertz radiation is notoriously difficult(2). The hot-electron photothermoelectric effect in graphene is a promising detection mechanism; photoexcited carriers rapidly thermalize due to strong electron-electron interactions(3,4), but lose energy to the lattice more slowly(3,5). The electron temperature gradient drives electron diffusion, and asymmetry due to local gating(6,7) or dissimilar contact metals(8) produces a net current via the thermoelectric effect. Here, we demonstrate a graphene thermoelectric terahertz photodetector with sensitivity exceeding 10 V W-1 (700 V W-1) at room temperature and noise-equivalent power less than 1,100 pW Hz(-1/2) (20 pW Hz(-1/2)), referenced to the incident (absorbed) power. This implies a performance that is competitive with the best room-temperature terahertz detectors(9) for an optimally coupled device, and time-resolved measurements indicate that our graphene detector is eight to nine orders of magnitude faster than those(7,10). A simple model of the response, including contact asymmetries (resistance, work function and Fermi-energy pinning) reproduces the qualitative features of the data, and indicates that orders-of-magnitude sensitivity improvements are possible.