It is now generally accepted that galaxies host supermassive black holes in their centres. Observational evidence for a correlation between the supermassive black hole and the host galaxy properties are numerous, such as the relation between the mass of the black hole and the mass of the galaxy's bulge (MBH-Mbulge relation) and between the black hole mass and the velocity dispersion of a galaxy (MBH-σ relation). However, attempts at correlating the growth of galaxies (star formation rate - SFR) with the growth of their supermassive black holes (black hole accretion rate - BHAR) yield conflicting results, depending on the properties of the selected sample. Furthermore, most samples used when studying the activity of SMBHs are taken from AGN-selected populations, which may introduce biases by restricting the studies to only high luminosity/high accreting sources. This thesis takes an alternative approach and probes the relation between the SFR and BHAR for samples of star-forming galaxies across cosmic time from z~0.4 to z~6. It makes use of the High Redshift Emission Line Survey (HiZELS) and the Slicing COSMOS with 4000 Lyman-α emitters (SC4K survey) to select a large sample of line-emitting star-forming galaxies (672 Hα emitters - HAEs - at z=0.4-2 for HiZELS and 3700 Lyman-α emitters - LAEs - at z=2-6 for SC4K). Making use of stacking and SED fitting techniques, as well as direct source extraction by using the publicly available data in the X-rays, radio and FIR bands for the COSMOS field, this work estimates the BHARs and SFRs of star-forming galaxies and provides additional information on the SMBH/SF processes of star-forming galaxies across cosmic time. The results show that most star-forming galaxies at z3σ) when stacking in the radio while excluding direct detections, allowing for the use of radio as an additional SFR estimator. The results from radio are found to be consistent with FIR and Lyα results (total median SFR~7.2 Msun y-1). The BHAR/SFR ratio of LAEs (log10(BHAR/SFR)<-2.7) is comparable to that of lower redshift HAEs (log10(BHAR/SFR)=-3.3) and sets a trend where star-forming galaxies grow approximately 1000 times faster than their SMBHs. This thesis results are therefore consistent with a scenario of co-evolution between supermassive black holes and their host galaxies
