We present a comprehensive X-ray study of the population of supernova remnants (SNRs) in the LMC. Using primarily XMM-Newton, we conduct a systematic spectral analysis of LMC SNRs to gain new insights on their evolution and the interplay with their host galaxy. We combined all the archival XMM observations of the LMC with those of our Very Large Programme survey. We produced X-ray images and spectra of 51 SNRs, out of a list of 59. Using a careful modelling of the background, we consistently analysed all the X-ray spectra and measure temperatures, luminosities, and chemical compositions. We investigated the spatial distribution of SNRs in the LMC and the connection with their environment, characterised by various SFHs. We tentatively typed all LMC SNRs to constrain the ratio of core-collapse to type Ia SN rates in the LMC. We compared the X-ray-derived column densities to HI maps to probe the three-dimensional structure of the LMC. This work provides the first homogeneous catalogue of X-ray spectral properties of LMC SNRs. It offers a complete census of LMC SNRs exhibiting Fe K lines (13% of the sample), or revealing contribution from hot SN ejecta (39%). Abundances in the LMC ISM are found to be 0.2-0.5 solar, with a lower [$alpha$/Fe] than in the Milky Way. The ratio of CC/type Ia SN in the LMC is $N_{mathrm{CC}}/N_{mathrm{Ia}} = 1.35(_{-0.24}^{+0.11})$, lower than in local SN surveys and galaxy clusters. Comparison of X-ray luminosity functions of SNRs in Local Group galaxies reveals an intriguing excess of bright objects in the LMC. 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. (abridged)
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