<p>With the advent of time-domain astronomy and the game-changing next generation of telescopes, we have unprecedented opportunities to explore the most energetic events in our Universe through electromagnetic radiation, gravitational waves and neutrinos. These are elementary particles, which exist in three different flavours and change the latter as they propagate in the dense core of astrophysical sources as well as en route to Earth. To capitalize on existing and upcoming multi-messenger opportunities, it is crucial to understand: (1) the role of neutrinos in explosive transient sources as well as in the synthesis of the elements heavier than iron; (2) the impact of neutrino physics on the multi-messenger observables and (3) the information on the source physics carried by the detectable neutrino signal. In this Review, the status of this exciting and fast-moving field is outlined, focusing on astrophysical sources linked to collapsing massive stars and neutron-star mergers. In the light of the upcoming plethora of multi-messenger data, outstanding open issues concerning the optimization of multi-messenger detection strategies are discussed.</p>