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We describe system verification tests and early science results from the pulsar processor (PTUSE) developed for the newly-commissioned 64-dish SARAO MeerKAT radio telescope in South Africa. MeerKAT is a high-gain (~2.8 K/Jy) low-system temperature (~18 K at 20cm) radio array that currently operates from 580-1670 MHz and can produce tied-array beams suitable for pulsar observations. This paper presents results from the MeerTime Large Survey Project and commissioning tests with PTUSE. Highlights include observations of the double pulsar J0737-3039A, pulse profiles from 34 millisecond pulsars from a single 2.5h observation of the Globular cluster Terzan 5, the rotation measure of Ter5O, a 420-sigma giant pulse from the Large Magellanic Cloud pulsar PSR J0540-6919, and nulling identified in the slow pulsar PSR J0633-2015. One of the key design specifications for MeerKAT was absolute timing errors of less than 5 ns using their novel precise time system. Our timing of two bright millisecond pulsars confirm that MeerKAT delivers exceptional timing. PSR J2241-5236 exhibits a jitter limit of <4 ns per hour whilst timing of PSR J1909-3744 over almost 11 months yields an rms residual of 66 ns with only 4 min integrations. Our results confirm that the MeerKAT is an exceptional pulsar telescope. The array can be split into four separate sub-arrays to time over 1000 pulsars per day and the future deployment of S-band (1750-3500 MHz) receivers will further enhance its capabilities.