A number of theoretical mechanisms have been invoked to explain the transformation of star forming galaxies into quenched galaxies, such as the removal of gas by active galactic nuclei- (AGN) and star formation-driven galactic outflows, and the heating of the galaxy halo and subsequent halting of gas accretion caused by virial shocks and preventative AGN feedback. Nonetheless, we still lack a clear understanding of which mechanism is the most important for quenching galaxies. The goal of this thesis is to study galaxy quenching from an observational perspective and to address the following question: why do galaxies stop forming stars?

In the first part of this thesis I examine the relationship between molecular gas (observed with ALMA and NOEMA) and star formation on spatially resolved scales in seven green valley MaNGA galaxies. I find that both the star formation efficiency (SFE) and gas fraction of green valley galaxies is suppressed relative to typical star forming galaxies. These two effects are equally important for suppressing star formation in the outer disc, but it is not possible to rank their effect towards the strongly quenched galaxy centre, where SFE is only constrained to an upper limit. I also show that the significant reduction in gas fraction in the central regions is driven by an increase in stellar mass surface density rather than a decrease in molecular gas surface density, which disfavours quenching mechanisms that eject gas from galaxies.

In the second part of this thesis I develop a 2D inclined rotating disc model to estimate the kinematic properties of 1862 MaNGA galaxies, and I use a random forest analysis to explore the relationship between galaxy kinematics and quenching. I find that the average velocity dispersion is overwhelmingly the most important kinematic parameter for predicting galaxy quenching. Furthermore, a full and partial correlation analysis shows that many commonly discussed correlations between galaxy properties (such as stellar mass) and quenching are spurious, and indeed at fixed velocity dispersion these correlations are almost entirely removed. I interpret this finding in the context of the BH-σ relation and argue for a scenario in which quenching occurs due to preventative feedback from AGN.

In the third part of this thesis I attempt to directly probe preventative feedback from AGN by analysing a deep ALMA band 4 observation of HE0515-4414 to search for the Sunyaev-Zel'dovich (SZ) signal tracing the quasar's hot halo gas. I find marginal evidence (~3.3σ) of large scale (~300 kpc) heating in the visibility plane, as well as ~3.2σ evidence of localised heating to the south west of the central quasar in the image plane. Furthermore, I use SZ maps from the FABLE simulation to demonstrate that ALMA and the ACA are effective at detecting the SZ signal in band 3, and I discuss the optimal strategy for future observations.