The MeerKAT telescope represents an outstanding opportunity for radio pulsar timing science with its unique combination of a large collecting area and aperture efficiency (effective area $\sim$7500 m$^2$), system temperature ($T<20$K), high slew speeds (1-2 deg/s), large bandwidths (770 MHz at 20cm wavelengths), southern hemisphere location (latitude $\sim -30^\circ$) and ability to form up to four sub-arrays. The MeerTime project is a five-year program on the MeerKAT array by an international consortium that will regularly time over 1000 radio pulsars to perform tests of relativistic gravity, search for the gravitational wave signature induced by supermassive black hole binaries in the timing residuals of millisecond pulsars, explore the interiors of neutron stars through a pulsar glitch monitoring programme, explore the origin and evolution of binary pulsars, monitor the swarms of pulsars that inhabit globular clusters and monitor radio magnetars. MeerTime will complement the TRAPUM project and time pulsars TRAPUM discovers in surveys of the galactic plane, globular clusters and the galactic centre. In addition to these primary programmes, over 1000 pulsars will have their arrival times monitored and the data made immediately public. The MeerTime pulsar backend comprises two server-class machines each of which possess four Graphics Processing Units. Up to four pulsars can be coherently dedispersed simultaneously up to dispersion measures of over 1000 pc cm$^{-3}$. All data will be provided in psrfits format. The MeerTime backend will be capable of producing coherently dedispersed filterbank data for timing multiple pulsars in the cores of globular clusters that is useful for pulsar searches of tied array beams. The first real-time pulsar profiles have been obtained as part of the MeerKAT commissioning process, and useful scientific data will start to come online through 2017. All MeerTime data will ultimately be made available for public use, and any published results will include the arrival times and profiles used in the results.