Cosmic explosions dissipate energy into their surroundings on a very wide range of time scales: producing shock waves and associated particle acceleration. The historical culprits for the acceleration of the bulk of Galactic cosmic rays are supernova remnants: explosions on approximately 10 4 year time scales. Increasingly, however, time-variable emission points to rapid and efficient particle acceleration in a range of different astrophysical systems. Gamma-ray bursts have the shortest time scales, with inferred bulk Lorentz factors of approximately 1000 and photons emitted beyond 100 GeV, but active galaxies, pulsar wind nebulae and colliding stellar winds are all now associated with time-variable emission at approximately teraelectron volt energies. Cosmic photons and neutrinos at these energies offer a powerful probe of the underlying physical mechanisms of cosmic explosions, and a tool for exploring fundamental physics with these systems. Here, we discuss the motivations for high-energy observations of transients, the current experimental situation, and the prospects for the next decade, with particular reference to the major next-generation high-energy observatory, the Cherenkov Telescope Array.