Star-formation history will become quantifiable with increasing accuracy using recent and forthcoming instruments. The evolution can be traced back from well known present-day galaxies, or directly observed for galaxies at different look-back times. This requires clear and consistently matched selection criteria in the galaxy selection, the best of these, perhaps, being to limit galaxy fluxes at wavelengths representative of their bolometric luminosities. Star-formation diagnostics are discussed, as well as their availability for redshifted galaxies. The derived star-formation rates (SFRs) can be affected by extinction and by interstellar-medium (ISM) properties, by model uncertainties and also by contamination of active galactic nuclei (AGN) and of old starlight. The decrease of SFR comoving density, at least since z similar to 1, appears to be corroborated by many such diagnostics. Apparently large disks are not contributing consistently to the observed decrease, which is mainly related to significant changes with epoch in the distribution of galaxy morphologies. A significant fraction of the global star formation occurs in a relatively small number of obscured galaxies, often seen in interacting systems.