We present a comprehensive X-ray study of the population of supernova remnants (SNRs) in the Large Magellanic Cloud (LMC). Using primarily XMM-Newton observations, we conduct a systematic spectral analysis of LMC SNRs to gain new insight into their evolution and the interplay with their host galaxy. We combined all the archival XMM-Newton observations of the LMC with those of our Very Large Programme LMC survey. We produced X-ray images and spectra of 51 SNRs, out of a list of 59 objects compiled from the literature and augmented with newly found objects. Using a careful modelling of the background, we consistently analysed all the X-ray spectra and measure temperatures, luminosities, and chemical compositions. The locations of SNRs are compared to the distributions of stars, cold gas, and warm gas in the LMC, and we investigated the connection between the SNRs and their local environment, characterised by various star formation histories. We tentatively typed all LMC SNRs, in order to constrain the ratio of core-collapse to type Ia SN rates in the LMC. We also compared the column densities derived from X-ray spectra to HI maps, thus probing the three-dimensional structure of the LMC. This work provides the first homogeneous catalogue of the X-ray spectral properties of SNRs in the LMC. It offers a complete census of LMC remnants whose X-ray emission exhibits Fe K lines (13% of the sample), or reveals the contribution from hot supernova ejecta (39%), which both give clues to the progenitor types. The abundances of O, Ne, Mg, Si, and Fe in the hot phase of the LMC interstellar medium are found to be between 0.2 and 0.5 times the solar values with a lower abundance ratio [{alpha}/Fe] than in the Milky Way. The current ratio of core-collapse to type Ia SN rates in the LMC is constrained to N_CC_/N_Ia_=1.35(_-0.24_^+0.11^), which is lower than in local SN surveys and galaxy clusters. Our comparison of the X-ray luminosity functions of SNRs in Local Group galaxies (LMC, SMC, M31, and M33) reveals an intriguing excess of bright objects in the LMC. Finally, we confirm that 30 Doradus and the LMC Bar are offset from the main disc of the LMC to the far and near sides, respectively.