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We present precision radial velocities and stellar population parameters for 77 star clusters in the Local Group galaxy M33. Our Gran Telescopio de Canarias and William Herschel Telescope observations sample both young, massive clusters and known/candidate globular clusters (GCs), spanning ages ~106-1010 yr, and metallicities, [M/H] ~ -1.7 to solar. The cluster system exhibits an age-metallicity relation; the youngest clusters are the most metal rich. When compared to HI data, clusters with [M/H] ~ -1.0 and younger than ~4 Gyr are clearly identified as a disc population. The clusters show evidence for strong time evolution in the disc radial metallicity gradient (d[M/H]dt/dR = 0.03 dex kpc-1 Gyr-1). The oldest clusters have stronger, more negative gradients than the youngest clusters in M33. The clusters also show a clear age-velocity dispersion relation. The line-of-sight velocity dispersions of the clusters increases with age similar to Milky Way open clusters and stars. The general shape of the relation is reproduced by disc heating simulations, and the similarity between the relations in M33 and the Milky Way suggests that heating by substructure and cooling of the interstellar medium both play a role in shaping this relation. We identify 12 'classical' GCs, six of which are newly identified GC candidates. The GCs are more metal rich than Milky Way halo clusters, and show weak rotation. The inner (R < 4.5 kpc) GCs exhibit a steep radial metallicity gradient (d[M/H]/dR = -0.29 ± 0.11 dex kpc-1) and an exponential-like surface density profile. We argue that these inner GCs are thick disc rather than halo objects. [ABSTRACT FROM AUTHOR]