Over the last few decades a broad picture has been established of how stars of different masses end the lives: low- and intermediate-mass stars lose a large fraction of their initial mass and become white dwarfs, possibly connected with the formation of a planetary nebula; massive stars either produce neutron stars or black holes, mostly, but not always  associated with a supernova explosion; and the most massive stars may even disrupt themselves completely. I will first discuss the basic theoretical framework to understand these different fates and some of the empirical, observational evidence supporting it. I will then discuss some key fundamental questions that are not properly understood to date, specifically what causes the extreme mass loss of low- and intermediate-mass stars that ultimately terminates their evolution, and what determines the core collapse fate of massive stars, both areas in which there has been dramatic progress in recent year. I will also show how gravitational-wave observations with LIGO can be used to test some of these predictions in the very near future. Those unable to attend the colloquium in person are invited to participate online through Zoom (Meeting ID: 942 0262 2849, passcode 792771) using the link: https://eu02web.zoom-x.de/j/94202622849?pwd=dGlPQXBiUytzY1M2UE5oUDRhbzNOZz09 Professor Podsiadlowski is available for meetings by arrangement with his hosts, Friedrich Roepke (roepke@uni-heidelberg.de) and Fabian Schneider (fabian.schneider@h-its.org).