Abstract Experimental evidence in favor of or against the existence of a universal mechanism for the acceleration of charged particles in the Universe is analyzed from the modern standpoint. We adopt a purely phenomenological approach to this very intricate problem, discussing spatial scales that range from the magnetospheres of Earth and other planets of the Solar System to the Sun’s atmosphere, the heliosphere, supernovae, and extragalactic objects responsible for the generation of ultra-high-energy cosmic rays. We demonstrate a great variety of acceleration mechanisms operating in the Universe. However, data on the nuclear composition of accelerated particles obtained in numerous experiments may be indicative of a global nature of the Fermi-type stochastic acceleration mechanism inherent in various astrophysical objects. This mechanism may well be dominant over others, but a number of experimental observations are nevertheless inconsistent with that idea. It is possible that a hierarchy of accelerating mechanisms operates in some space objects, with preliminary acceleration by one mechanism followed by other mechanisms, acting sequentially or alternatively. We therefore discuss all currently available data for and against the global ‘presence’ of a Fermi-type acceleration mechanism.