PEBBLES is an ultra-deep continuum survey of the circumstellar disks that are predicted to be the most conducive to planet formation. Imaging the thermal emission from pebble-sized dust grains will show where and when planet-core growth is proceedings, and identify actual accreting proto-planets. The survey sample comprises a mass-limited cut from all known northern disks with long-millimetre wavelength dust emission, above a threshold of 2.5 times the minimum-mass Solar-nebula, at the theoretical boundary for forming the Sun's planets. The sample is otherwise unbiased and includes 19 young stellar objects and imageable disks in 13 fields, at distances <250 pc so that at 40 mas resolution the terrestrial planet zone is separated from that of gas giant formation. The span of stellar ages is ~0.1-7 Myr, i.e. the epochs of gas giant growth and early assembly stages of terrestrial planets. All systems will be imaged at C-band (5 cm) to a uniform mass depth of a few M_earth of dust (in the beam containing the Earth-formation zone requiring 468 hours in total including the Lovell Telescope. This will be the first survey of the inner disk regions at a few AU resolution and will exploit the uniquely optical-thin flux from these zones of very high column density. The survey results will show how planet growth proceeds - where, when, and with what outcomes - for comparison to inferred histories of the Sun and extrasolar planetary systems, and to our simulation results based on current planet-formation models. The scientfic legacy will also include measuring quantities vital to theoretical progress - particle sizes, disk surface densities and radial distributions, for the first time on few-AU scales - and providing a database of proto-planet targets for future followup with EVLA, ALMA and SKA.