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 } ) – star formation rate ( SFR ) plane with \log ( M _ { \star } / M _ { \odot } ) = 10 - 11.8 and SFR = 3.5 - 500 ~ { } M _ { \odot } 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 2.10 ^ { 9 } and 5.10 ^ { 10 } ~ { } M _ { \odot } 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 \widetilde { \mu _ { gas } } = 0.28 \pm 0.04 , gas fraction \widetilde { f _ { gas } } = 0.22 \pm 0.02 , and depletion time \widetilde { t _ { depl } } = 0.84 \pm 0.07 ~ { } 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 .