Three interrelated problems are discussed in this paper-the theory of the extragalactic origin of cosmic rays, the origin of the X-ray background and the evolution of extragalactic radio sources. Cosmic ray electrons in extragalactic radio sources produce not only radio waves by the synchrotron mechanism but also X-rays by inverse Compton scattering of the relict radiation. The latter is likely to be the dominant loss mechanism for electrons at large redshifts. There then exists a simple relation between the local flux of cosmic rays and the X-ray background. Using the conventional value = ice for the energy densities in relativistic protons and electrons generated in radio sources, the upper limit to the local energy density of cosmic rays due to extragalactic radio sources is ic-3 eV . Only if > ic4 can the whole of the observed cosmic rays be attributed to extragalactic sources. Although the total energy associated with the X-ray background can be produced in this model, there are very severe difficulties in accounting for the break in the observed background spectrum at Ex 40 keV. These arise because the available radio data give information about the injection spectra of cosmic ray electrons into the intergalactic medium. It is also shown that the apparent cut-off in the distribution of radio sources at large redshifts cannot be attributed to the effects of inverse Compton scattering.