We introduce the BEBOP radial velocity survey for circumbinary planets. We initiated this survey using the CORALIE spectrograph on the Swiss Euler Telescope at La Silla, Chile. An intensive four-year observation campaign commenced in 2013, targeting 47 single-lined eclipsing binaries drawn from the EBLM survey for low mass eclipsing binaries. Our specific use of binaries with faint M dwarf companions avoids spectral contamination, providing observing conditions akin to single stars. By combining new BEBOP observations with existing ones from the EBLM programme, we report on the results of 1519 radial velocity measurements over timespans as long as eight years. For the best targets we are sensitive to planets down to 0.1 M_Jup, and our median sensitivity is 0.4 M_Jup. In this initial survey we do not detect any planetary mass companions. Nonetheless, we present the first constraints on the abundance of circumbinary companions, as a function of mass and period. A comparison of our results to Kepler’s detections indicates a dispersion of planetary orbital inclinations less than ~10°.