Abstract We present the new TNG50 cosmological, magnetohydrodynamical simulation – the third and final volume of the IllustrisTNG project. This simulation occupies a unique combination of large volume and high resolution, with a 50 Mpc box sampled by 21603 gas cells (baryon mass of 8 × 104 M⊙). The median spatial resolution of star-forming interstellar medium gas is ∼100−140 pc. This resolution approaches or exceeds that of modern ‘zoom’ simulations of individual massive galaxies, while the volume contains ∼20 000 resolved galaxies with $M_⋆ \gtrsim 10^7$ M⊙. Herein we show first results from TNG50, focusing on galactic outflows driven by supernovae as well as supermassive black hole feedback. We find that the outflow mass loading is a non-monotonic function of galaxy stellar mass, turning over and rising rapidly above 1010.5 M⊙ due to the action of the central black hole (BH). The outflow velocity increases with stellar mass, and at fixed mass it is faster at higher redshift. The TNG model can produce high-velocity, multiphase outflows that include cool, dense components. These outflows reach speeds in excess of 3000 km s−1 out to 20 kpc with an ejective, BH-driven origin. Critically, we show how the relative simplicity of model inputs (and scalings) at the injection scale produces complex behaviour at galactic and halo scales. For example, despite isotropic wind launching, outflows exhibit natural collimation and an emergent bipolarity. Furthermore, galaxies above the star-forming main sequence drive faster outflows, although this correlation inverts at high mass with the onset of quenching, whereby low-luminosity, slowly accreting, massive BHs drive the strongest outflows.