  Active galactic nuclei (AGN) are believed to regulate star formation inside
their host galaxies through "AGN feedback". We summarise our on-going study of
luminous AGN (z~0.2-3; L_(AGN,bol)>~10^43 erg/s), which is designed to search
for observational signatures of feedback by combining observed star-formation
rate (SFR) measurements from statistical samples with cosmological model
predictions. Using the EAGLE hydrodynamical cosmological simulations, in
combination with our Herschel+ALMA surveys, we show that - even in the presence
of AGN feedback - we do not necessarily expect to see any relationships between
average galaxy-wide SFRs and instantaneous AGN luminosities. We caution that
the correlation with stellar mass for both SFR and AGN luminosity can
contribute to apparent observed positive trends between these two quantities.
On the other hand, the EAGLE simulations, which reproduce our observations,
predict that a signature of AGN feedback can be seen in the wide specific SFR
distributions of $all$ massive galaxies (not just AGN hosts). Overall, whilst
we can not rule out that AGN have an immediate small-scale impact on in-situ
star-formation, all of our results are consistent with a feedback model where
galaxy-wide in-situ star formation is not rapidly suppressed by AGN, but where
the feedback likely acts over a longer timescale than a single AGN episode.
