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We study the formation of ultra-diffuse galaxies (UDGs) using the cosmological hydrodynamical simulation TNG50 of the Illustris-TNG suite. We define UDGs as dwarf galaxies in the stellar mass range $\rm{7.5 \leq log (M_{\star} / M_{\odot}) \leq 9 }$ that are in the $5\%$ most extended tail of the simulated mass-size relation. This results in a sample of UDGs with half-mass radii $\rm{r_{h \star } \gtrsim 2 \ kpc}$ and surface brightness between $\rm{24.5}$ and $\rm{28 \ mag \ arcsec^{-2}}$, similar to definitions of UDGs in observations. The large cosmological volume in TNG50 allows for a comparison of UDGs properties in different environments, from the field to galaxy clusters with virial mass $\rm{M_{200} \sim 2 \times 10^{14} ~ M_{\odot}}$. All UDGs in our sample have dwarf-mass haloes ($\rm{M_{200}\sim 10^{11} ~ M_{\odot} }$) and show the same environmental trends as normal dwarfs: field UDGs are star-forming and blue while satellite UDGs are typically quiescent and red. The TNG50 simulation predicts UDGs that populate preferentially higher spin haloes and more massive haloes at fixed $\rm{M_{\star}}$ compared to non-UDG dwarfs. This applies also to most satellite UDGs, which are actually ``born" UDGs in the field and infall into groups and clusters without significant changes to their size. We find, however, a small subset of satellite UDGs ($\lesssim 10 \%$) with present-day stellar size a factor $\geq 1.5$ larger than at infall, confirming that tidal effects, particularly in the lower mass dwarfs, are also a viable formation mechanism for some of these dwarfs, although subdominant in this simulation.