In this review much of the current knowledge about the physics of anomalous cosmic rays is surveyed. These energetic particles are, most probably, accelerated at the so-called heliospheric shock which terminates the supersonic solar wind expansion. After a presentation of the general scenario embracing a description of the heliosphere. the basic paradigm explaining the existence of anomalous cosmic rays and the relevant observations, the main problems connected to the physics of this particular cosmic-ray component are identified and discussed in detail. To this end, the characteristics of the progenitor as well as descendant particle populations, i.e. interstellar neutral atoms. pick-up ions, and energetic neutral atoms. and those of galactic cosmic rays are described from an observational as well as a theoretical perspective, as far as they are related to heliospheric research. The relevance of heliospheric physics is pointed out as a vital link between basic (plasma) physics and (extra-heliospheric) astrophysics as well as a test bed for astrophysical concepts.