We present the PHANGS-MUSE survey, a programme that uses the MUSE integral field spectrograph at the ESO VLT to map 19 massive (9.4 <  log(M⋆/M⊙)< 11.0) nearby (D ≲ 20 Mpc) star-forming disc galaxies. The survey consists of 168 MUSE pointings (1′ by 1′ each) and a total of nearly 15 × 106 spectra, covering ∼1.5 × 106 independent spectra. PHANGS-MUSE provides the first integral field spectrograph view of star formation across different local environments (including galaxy centres, bars, and spiral arms) in external galaxies at a median resolution of 50 pc, better than the mean inter-cloud distance in the ionised interstellar medium. This ‘cloud-scale’ resolution allows detailed demographics and characterisations of H II regions and other ionised nebulae. PHANGS-MUSE further delivers a unique view on the associated gas and stellar kinematics and provides constraints on the star-formation history. The PHANGS-MUSE survey is complemented by dedicated ALMA CO(2–1) and multi-band HST observations, therefore allowing us to probe the key stages of the star-formation process from molecular clouds to H II regions and star clusters. This paper describes the scientific motivation, sample selection, observational strategy, data reduction, and analysis process of the PHANGS-MUSE survey. We present our bespoke automated data-reduction framework, which is built on the reduction recipes provided by ESO but additionally allows for mosaicking and homogenisation of the point spread function. We further present a detailed quality assessment and a brief illustration of the potential scientific applications of the large set of PHANGS-MUSE data products generated by our data analysis framework. The data cubes and analysis data products described in this paper represent the basis for the first PHANGS-MUSE public data release and are available in the ESO archive and via the Canadian Astronomy Data Centre.