ALMA has been working now for about 7 years since its commissioning in 2011, and has acccumulated a wealth of data, which have yet to be digested and interpreted. In many domains, ALMA has provided impressive breakthroughs, with unprecedented sensitivity and spatial resolution.
One of the main goals in galaxy evolution is to determine the molecular gas properties of galaxies as a function of redshift, to better understand the cosmic star formation history. This has been done through pointed observations, with large sample of objects selected from their stellar mass and SFR, being on the main sequence of star forming galaxies, where are born 90% of the stars in the Universe. Two main factors have been emphasized: the gas fraction increases steadily on the main sequence, as (1 + z)2, and this is the main reason of the peak in the SFRD at z ∼ 2. At a lower level, the star formation efficiency is also increasing with redshift, as (1 + z)α, with α = 0.6-1, (or the depletion time is decreasing, as (1 + z)−α), although the cause of this has not be clearly identified: either due to smaller and denser galaxies, with a shorter dynamical time, or due to a larger importance of galaxy interactions. Although some galaxy molecular maps have been done, and resolved Kennicutt-Schmidt relation explored, this is only the beginning, and the influence of morphology, kinematics and dynamics of galaxies is not yet understood.
In parallel, deep blind surveys, completely unbiased by previous wavelengths, have been carried out focussed either on the dust continuum, or the CO lines, with shallow or deeper approaches, according to the surface covered. The hope was to detect dusty galaxies at very high redshift, not suspected by other surveys. Eventually, dusty and massive galaxies are not as frequent as previously hoped at z larger than 5, which confirms the high-z drop in optical surveys. ALMA is now opening clearly the windows of the epoch of re-ionization, and it is likely that the main actors to reionize the Universe will turn out to be a large number of small and dwarf galaxies, while the major starbursts and quasars have a minor influence.
ALMA surveys have begun to unveil the cosmic evolution of the H2 content, but this is only the beginning, with huge error-bars, not allowing to disentangle the various theoretical models, as shown in Figure 8. In the future, this cosmic evolution will be compared with the star formation history, and also with the atomic gas content, to have a more precise budget of gas and star formation at all epochs.
There remain a large number of unsolved issues, like the symbiotic evolution of black hole and bulges in galaxies, which appear to be divergent at high redshift, the importance of AGN feedback in the early universe, the influence of environment in proto-clusters. The detection of important quantities of cold and dense gas in the circumgalactic medium at early times might give some clues in the missing baryon problem.
Acknowledgements I thank Paul Ho for inviting me to write this review.