Some stars cease to be in a bright and destructive display called a supernova. This thesis explores what we can learn about supernovae (SNe) by studying their immediate surroundings, and what the SNe can teach us about their environments. The work presented is mostly based on the rich harvest of observations from 2009-2017 by the Palomar Transient Factory (PTF) and its successor, the intermediate PTF (iPTF). The PTF/iPTF was an untargeted sky survey at Palomar Observatory, aimed at finding and following up astronomical transients, such as SNe. During its existence, a massive star typically loses several solar masses of material. If much mass is lost in the decades or centuries before the SN, this material around the star (the circumstellar medium, CSM) will be quickly swept up by the ejecta of the eventual SN. This interaction can contribute strongly to the luminosity of the SN and make the light curve of an interacting SN carry signs of the progenitor star mass loss history. SNe with a hydrogen-rich CSM are called SNe Type IIn. A SN of this type, iPTF13z, found and followed by iPTF, had a slowly declining lightcurve with at least 5 major rebrightenings ("bumps") indicating rich structure in the CSM. Archival images clearly shows a precursor outburst about 210 days before the SN discovery, demonstrating the iPTF13z progenitor to be restless before its demise. Type IIn supernovae are heterogeneous, but only limited statistics has been done on samples. From PTF/iPTF, a sample of 42 SNe Type IIn was therefore selected, with photometry allowing their light curve rise times, decline rates and peak luminosities to be measured. It was shown that more luminous events are generally more long-lasting, but no strong correlation was found between rise times and peak luminosities. Two clusters of risetimes (around 20 and 50 days, respectively) were identified. The less long-lasting SNe Type IIn dominate the sample, suggesting that stars with a less extended dense CSM might be more common among SN Type IIn progenitors. Thermonuclear SNe (SNe Type Ia) are useful as standardisable candles, but no secure identification has yet been made of the progenitor system of a SN Type Ia. Using a late-time spectrum from the Nordic Optical Telescope of the nearby thermonuclear SN 2014J, a search for material ablated from a possible non-compact companion gave the upper limit of about 0.0085 solar masses of hydrogen-rich ablated gas. One likely explanation is that the SN 2014J progenitor system was a binary white dwarf. Supernovae are also useful tracers of the star formation history in their host galaxies, with SNe Type Ia tracing earlier epochs of star formation and exploding massive stars tracing more recent. For active galactic nuclei (AGN, the luminous centres of galaxies harbouring accreting supermassive black holes) SNe allows the so-called unification model to be tested. The unification model assumes that the main distinction between the two types of AGN is the viewing angle towards the central black hole, and that other properties (e.g. star formation history) of the host galaxies should be the same for the two AGN types. Matching 2190 SNe from PTF/iPTF to about 89000 AGN with spectra from the Sloan Digital Sky Survey, a significantly higher number of SNe in the hosts of AGN type 2 was found, challenging the unification model.