Abstract We present a new cosmological, magnetohydrodynamical simulation for galaxy formation: TNG50, the third and final instalment of the IllustrisTNG project. TNG50 evolves 2 × 21603 dark matter particles and gas cells in a volume 50 comoving Mpc across. It hence reaches a numerical resolution typical of zoom-in simulations, with a baryonic element mass of $8.5\times 10^4\, {\rm M}_{⊙ }$ and an average cell size of 70–140 pc in the star-forming regions of galaxies. Simultaneously, TNG50 samples ∼700 (6500) galaxies with stellar masses above $10^{10} \, (10^8)\, {\rm M}_{⊙ }$ at $z$ = 1. Here we investigate the structural and kinematical evolution of star-forming galaxies across cosmic time (0 ≲ $z$ ≲ 6). We quantify their sizes, disc heights, 3D shapes, and degree of rotational versus dispersion-supported motions as traced by rest-frame V-band light (i.e. roughly stellar mass) and by $\rm H\,α$ light (i.e. star-forming and dense gas). The unprecedented resolution of TNG50 enables us to model galaxies with sub-kpc half-light radii and with ≲300-pc disc heights. Coupled with the large-volume statistics, we characterize a diverse, redshift- and mass-dependent structural and kinematical morphological mix of galaxies all the way to early epochs. Our model predicts that for star-forming galaxies the fraction of disc-like morphologies, based on 3D stellar shapes, increases with both cosmic time and galaxy stellar mass. Gas kinematics reveal that the vast majority of $10^{9-11.5}\, {\rm M}_{⊙ }$ star-forming galaxies are rotationally supported discs for most cosmic epochs (Vrot/σ > 2–3, $z$ ≲ 5), being dynamically hotter at earlier epochs ($z$ ≳ 1.5). Despite large velocity dispersion at high redshift, cold and dense gas in galaxies predominantly arranges in disky or elongated shapes at all times and masses; these gaseous components exhibit rotationally dominated motions far exceeding the collisionless stellar bodies.