Acceleration of energetic particles is common to almost all astrophysical plasmas, e.g. galactic cosmic rays in the interstellar medium, energetic particles in solar flares. There is also a ubiquitous presence of suprathermal particles in the solar wind. Two mechanisms are usually invoked to explain the acceleration: stochastic acceleration, which is relatively slow, and diffusive shock acceleration, which is the usual mechanism of choice. However, recent observations from Voyager in the heliosheath, as well as observations in the inner heliosphere, have challenged whether either of these mechanisms are dominant. The observed spectra of accelerated particles in disparate plasma conditions has a common spectral shape: a power law in particle speed with spectral index of -5 when expressed as a distribution function, or equivalently, a power law in particle energy with spectral index of -1.5 when expressed as differential intensity. Neither diffusive shock acceleration nor stochastic acceleration can explain the common spectral shape. A new acceleration process has been developed: a pump acceleration that relies on the interplay between acceleration in compression regions and escape by spatial diffusion. The mechanism can account for all the heliospheric observations. Some of the subtleties of the pump acceleration process will be discussed, and it will be applied to explain the acceleration of galactic cosmic rays in the interstellar medium.