We propose that star-forming galaxies are a major, perhaps dominant, component of the X-ray background (∼2-10 keV). Such star-forming galaxies may be largely powered by superluminous Population I massive X-ray binaries (MXRB), formed in the wake of star formation in regions of low metallicity. MXRB have very hard X-ray spectra (2-30 keV) thus matching the overall spectrum of the XRB extremely well after subtraction of the contribution from Seyferts and quasars and allowing for redshifts of the star-forming galaxies up to at least ∼ 1. Star-forming galaxies at moderate redshift and with moderate numbers (100-1000) of MXRB may evolve into the infrared galaxies found at low redshifts using IRAS, and may also be closely related to those galaxies identified with sub-mJy radio sources. We find that the relationship between X-ray and infrared luminosities for starburst/peculiar galaxies and IRAS galaxies is indistinguishable. Using this correlation to convert the IRAS galaxy luminosity function to an X-ray luminosity function, we estimate a contribution to the XRB at 2 keV in the range 20%-30% with moderate evolution, rising to at least 50% with evolution similar to that of active galactic nuclei (AGN). In the energy range ∼2-10 keV, star-forming galaxies at intermediate redshifts may dominate the XRB. We also predict the X-ray number counts of star-forming galaxies, showing that one needs to reach very low fluxes ∼ 10-15-10-16 ergs cm-2 s-1 to quantify directly any X-ray evolution.