Observations of the cosmic evolution of different gas phases across time indicate a marked increase in the molecular gas mass density towards z ∼ 2 − 3. Such a transformation implies an accompanied change in the global distribution of molecular hydrogen column densities ($N_{\rm {H_2}}$). Using observations by PHANGS-ALMA/SDSS and simulations by GRIFFIN/IllustrisTNG we explore the evolution of this H2 column density distribution function [$f(N_{\rm {H}_2})$]. The H2 (and HI) column density maps for TNG50 and TNG100 are derived in post-processing and are made available through the IllustrisTNG online API. The shape and normalization of $f(N_{\rm {H}_2})$ of individual main-sequence star-forming galaxies are correlated with the star formation rate (SFR), stellar mass (M*), and H2 mass ($M_{\rm {H}_2}$) in both observations and simulations. TNG100, combined with H2 post-processing models, broadly reproduces observations, albeit with differences in slope and normalization. Also, an analytically modelled f(N), based on exponential gas disks, matches well with the simulations. The GRIFFIN simulation gives first indications that the slope of $f(N_{\rm {H}_2})$ might not majorly differ when including non-equilibrium chemistry in simulations. The $f(N_{\rm {H}_2})$ by TNG100 implies that higher molecular gas column densities are reached at z = 3 than at z = 0. Further, denser regions contribute more to the molecular mass density at z = 3. Finally, H2 starts dominating compared to HI only at column densities above log($N_{\rm {H}_2} / \rm {cm}^{-2}) \sim 21.8-22$ at both redshifts. These results imply that neutral atomic gas is an important contributor to the overall cold gas mass found in the ISM of galaxies including at densities typical for molecular clouds at z = 0 and z = 3.

The gas cycling in the circumgalactic regions of galaxies is known to be multi-phase. The MUSE-ALMA Haloes survey gathers a large multi-wavelength observational sample of absorption and emission data with the goal to significantly advance our understanding of the physical properties of such CGM gas. A key component of the MUSE-ALMA Haloes survey is the multi-facility observational campaign conducted with VLT/MUSE, ALMA and HST. MUSE-ALMA Haloes targets comprise 19 VLT/MUSE IFS quasar fields, including 32 zabs <0.85 strong absorbers with measured $N(\rm {H\,{\small I}})$ ≥1018 cm-2 from UV-spectroscopy. We additionally use a new complementary HST medium program to characterise the stellar content of the galaxies through a 40-orbit three-band UVIS and IR WFC3 imaging. Beyond the absorber-selected targets, we detect 3658 sources all fields combined, including 703 objects with spectroscopic redshifts. This galaxy-selected sample constitutes the main focus of the current paper. We have secured millimeter ALMA observations of some of the fields to probe the molecular gas properties of these objects. Here, we present the overall survey science goals, target selection, observational strategy, data processing and source identification of the full sample. Furthermore, we provide catalogues of magnitude measurements for all objects detected in VLT/MUSE, ALMA and HST broad-band images and associated spectroscopic redshifts derived from VLT/MUSE observations. Together, this data set provides robust characterisation of the neutral atomic gas, molecular gas and stars in the same objects resulting in the baryon census of condensed matter in complex galaxy structures.