An aqueous alkaline cyanide-free bath for the galvanic deposition of zinc or zinc alloy coatings on substrate surfaces is described, which contains in addition to a source of zinc ions and optionally a source of further metal ions, also hydroxide ions and a polymer of the general formula A that is soluble in the bath ##STR1## as well as optionally conventional additives. The bath may furthermore contain a quaternary derivative of a pyridine-3-carboxylic acid of the formula B and/or a quaternary derivative of a pyridine-3-carboxylic acid of the formula C ##STR2## A process for the galvanic deposition of zinc coatings and zinc alloy coatings using the aforedescribed bath is also described. By using the baths according to the invention it is possible to produce coatings that exhibit a uniform layer thickness combined with a high gloss, and which do not exhibit any tendency to exfoliate.