We discuss the influence of strong CP-violating effects in invisible-axion models on electroweak baryogenesis. We find that they can be important only if the electroweak phase transition (EPT) occurs simultaneously with the chirality-breaking phase transition in QCD. The predicted baryon asymmetry of the Universe (BAU) might be of order of the observable one only if there is no entropy production in the EPT. At the same time the dark-matter constraints put a lower bound on the temperature after the EPT, so that it is impossible to satisfy simultaneously both requirements (BAU and axion dark matter). We conclude that possible strong CP violation plays no role in electroweak baryogenesis, unless there exists some yet unknown mechanism for the BAU generation, producing baryons at a rate which has to be at least 3 orders of magnitude higher than the sphaleron-induced rate. In this kind of hypothetical scenario there is a prediction of the axionic dark matter in the Universe from the very existence of the baryon excess while the dark-matter density is generally of order or greater than that of the observed baryonic matter. © 1992 The American Physical Society.