We report on the preparation and characterization of epitaxial ACuO2 (A = Sr, Ca, Ba) thin films and multilayers with the so- called infinite layer (IL) structure, by rf magnetron sputtering. Films and multilayers without Ba have a remarkable crystal quality, whereas those containing this large ion are often multiphased and unstable. In spite of the excellent crystalline quality of these samples, obtaining thin films having both IL structure and displaying superconducting properties has not succeeded; our pure IL samples display semiconducting behavior, and the different procedures tried in order to dope them—annealings, introduction of disorder or cation vacancies, artificial layering—have failed. These results support that the pure IL structure ACuO2 (A = alkaline earth) cannot superconduct.