Nuclear activity and star-formation activity are two important processes that can shape galaxies. This thesis is dedicated to studying the most extreme nuclear activity and star-formation activity in our Universe. First I study the widely-debated topic about whether galaxy mergers can trigger active galactic nuclei (AGNs) and whether there exists difference of impact in triggering different types of AGNs. Then I study extreme dusty star-forming galaxies by selecting a parent sample of infrared (IR) luminous galaxies based on Herschel space telescope object catalogs. I propose a novel method to deblend Herschel imaging based on the tight correlation between radio and far-IR. Furthermore, I investigate the physical properties of the most extreme hyperluminous infrared galaxies in detail and their relation across cosmic time. Finally, I study the small-scale environment of these HLIRGs by searching for their neighbours and comparing with random positions.