The core collapse of massive stars and compact relativistic star mergers are accompanied by a rapid release of an enormous amount of energy, of the order of the rest energy of a star. Supernovae and gamma-ray bursts associated with these processes are observed almost every day by modern telescopes. Radiation from such sources is observed across the entire electromagnetic spectrum. Neutrinos from supernova 1987A and gravitational waves from relativistic star mergers have been detected. Along with rapidly variable and transient events, relativistic compact remnants of a collapsed star — accreting black holes and fast rotating pulsars — demonstrate high X-ray and gamma-ray luminosity for significantly longer times. The Crab Nebula and pulsars in gamma-ray binaries are excellent galactic laboratories that allow studying relativistic winds acting as cosmic high-energy particle accelerators. The study of physical processes leading to the conversion of the gravitational and rotational energy of relativistic objects into powerful electromagnetic radiation and high-energy neutrino fluxes provides unique opportunities for testing fundamental physical laws under extreme conditions unattainable in laboratory experiments on Earth. In this paper, we briefly review the results of observations and modeling of nonthermal processes in cosmic sources of high-energy radiation and discuss the prospects for advances in these studies.