We describe the ongoing Relativistic Binary programme (RelBin), a part of the MeerTime large survey project with the MeerKAT radio telescope. RelBin is primarily focused on observations of relativistic effects in binary pulsars to enable measurements of neutron star masses and tests of theories of gravity. We selected 25 pulsars as an initial high priority list of targets based on their characteristics and observational history with other telescopes. In this paper, we provide an outline of the programme, present polarisation calibrated pulse profiles for all selected pulsars as a reference catalogue along with updated dispersion measures. We report Faraday rotation measures for 24 pulsars, twelve of which have been measured for the first time. More than a third of our selected pulsars show a flat position angle swing confirming earlier observations. We demonstrate the ability of the Rotating Vector Model (RVM), fitted here to seven binary pulsars, including the Double Pulsar (PSR J0737$-$3039A), to obtain information about the orbital inclination angle. We present a high time resolution light curve of the eclipse of PSR J0737$-$3039A by the companions magnetosphere, a high-phase resolution position angle swing for PSR J1141$-$6545, an improved detection of the Shapiro delay of PSR J1811$-$2405, and pulse scattering measurements for PSRs J1227$-$6208, J1757$-$1854, and J1811$-$1736. Finally, we demonstrate that timing observations with MeerKAT improve on existing data sets by a factor of, typically, 2-3, sometimes by an order of magnitude.
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