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Following the success of the Plateau de Bure high-z Blue Sequence Survey (PHIBSS), we present the PHIBSS2 legacy program, a survey of the molecular gas properties of star-forming galaxies on and around the star-formation main sequence (MS) at different redshifts using IRAM's NOrthern Extended Millimeter Array (NOEMA). This survey significantly extends the existing sample of star-forming galaxies with CO molecular gas measurements, probing the peak epoch of star formation (z = 1 − 1.6) as well as its building-up (z = 2 − 3) and winding-down (z = 0.5 − 0.8) phases. The targets are drawn from the well-studied GOODS, COSMOS, and AEGIS cosmological deep fields and uniformly sample the MS in the stellar mass (M⋆) – star formation rate (SFR) plane with log(M⋆/M⊙) = 10 − 11.8 and SFR = 3.5 − 500 M⊙ yr−1 without morphological selection, thus providing a statistically meaningful census of star-forming galaxies at different epochs. We describe the survey strategy and sample selection before focusing on the results obtained at redshift z = 0.5 − 0.8, where we report 60 CO(2-1) detections out of 61 targets. We determine molecular gas masses between 2 × 109 and 5 × 1010 M⊙ and separately obtain disc sizes and bulge-to-total (B/T) luminosity ratios from HST I-band images. The median molecular gas-to-stellar mass ratio μgas∼ = 0.28 ± 0.04 μ gas ∼ = 0.28 ± 0.04 $ \widetilde{\mu_{\mathrm{gas}}} = 0.28 \pm 0.04 $ , gas fraction fgas∼ = 0.22 ± 0.02 f gas ∼ = 0.22 ± 0.02 $ \widetilde{f_{\mathrm{gas}}} = 0.22 \pm 0.02 $ , and depletion time t depl ∼ = 0.84 ± 0.07 Gyr $ \widetilde{t_{\mathrm{depl}}} = 0.84 \pm 0.07\,\mathrm{Gyr} $ as well as their dependence with stellar mass and offset from the MS follow published scaling relations for a much larger sample of galaxies spanning a significantly wider range of redshifts, the cosmic evolution of the SFR being mainly driven by that of the molecular gas fraction. The galaxy-averaged molecular Kennicutt–Schmidt (KS) relation between molecular gas and SFR surface densities is strikingly linear, pointing towards similar star formation timescales within galaxies at any given epoch. In terms of morphology, the molecular gas content, the SFR, the disc stellar mass, and the disc molecular gas fraction do not seem to correlate with B/T and the stellar surface density, which suggests an ongoing supply of fresh molecular gas to compensate for the build-up of the bulge. Our measurements do not yield any significant variation of the depletion time with B/T and hence no strong evidence for morphological quenching within the scatter of the MS. [ABSTRACT FROM AUTHOR]