We estimate the rate of the anomalous electroweak baryon-number non-conserving processes in the cosmic plasma and find that it exceeds the expansion rate of the universe at T > (a few) × 102 GeV. We study whether these processes wash out the baryon asymmetry of the universe (BAU) generated at some earlier state (say, at GUT temperatures). We also discuss the possibility of BAU generation by the electroweak processes themselves and find that this does not take place if the electroweak phase transition is of second order. No definite conclusion is made for the strongly first-order phase transition. We point out that the BAU might be attributed to the anomalous decays of heavy (MF>~ MW/αW) fermions if these decays are unsuppressed. On leave of absence from the Institute for Nuclear Research of the Academy of Sciences of the USSR, Moscow, USSR.