This thesis provides a comprehensive examination of star formation in the host galaxies of active galactic nuclei or AGN. AGN are bright, central regions of galaxies that are powered through accretion onto a supermassive black hole (SMBH). Through accretion and the loss of gravitational potential energy, AGN emit powerful radiation over all wavelengths of the electromagnetic spectrum. This radiation can influence the AGN's host galaxy through what is known as AGN ``feedback'' and is thought to suppress star formation as well as stop accretion onto the SMBH leading to a co-evolution between the SMBH and its host galaxy. Theoretical models have long invoked AGN feedback to be able reproduce the galaxy population we see today but observations have been unclear as to whether AGN actually have an effect on star formation.

To address this question, we selected a large sample of local ($z < 0.05$) AGN based on their detection at ultra-hard X-ray energies (14--195 keV) with the \textit{Swift} Burst Alert Telescope (BAT). Ultra-hard X-ray selection frees our sample from selection effects and biases due to obscuration and host galaxy contamination that can hinder other AGN samples. With these 313 BAT AGN we conducted a far-infrared survey using the \herschel \textit{Space Observatory}. We use the far-infrared imaging to probe the cold dust that traces recent star formation in the galaxy and construct spectral energy distributions (SEDs) from 12--500 \micron. We decompose the SEDs to remove the AGN contribution and measure infrared luminosity which provides us with robust estimates of the star formation rate (SFR).

Through a comparison with a stellar-mass matched non-AGN sample, we find that AGN host galaxies have larger dust masses, dust temperatures, and SFRs, confirming the results of previous studies that showed the optical colors of the BAT AGN are bluer than non-AGN. We find that the AGN luminosity as probed by the 14--195 keV luminosity is not related to the SFR of the host galaxy suggesting global, large scale star formation on an individual basis is not affected by the AGN. However, after a thorough analysis comparing our AGN to star-forming main sequence, a tight relationship between the SFR and stellar mass of a galaxy, we discover that our AGN as a whole show systematically lower specific SFRs (SFR/stellar mass). We confirm that AGN host galaxies, as a population, are transitioning between the star-forming and quiescent populations. This result supports the theory that AGN feedback has suppressed star formation, but we also consider other models that could reproduce our observations. Finally we conclude with a summary of this thesis and describe several ongoing and future projects that will push forward the exciting field of AGN research.