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The terminating layer of a perovskite (001) ABO₃ crystal, with A as an alkaline earth metal and B as a transition metal, influences the characteristics of the surface. The morphology during thermal treatment of the surface and epitaxial growth are determined by the composition of the surface, i.e., whether it is BO₂, AO, or a mixed-terminated surface. Since the morphology of a thin film is the result of the evolution of the surface during deposition, control from the very first layer is essential for smooth films. In the case of perovskite materials, this means control of the terminating layer. Here, we study the SrTiO₃ surface as a model system and the effects of the composition of the terminating layer, determined by the treatment and/or deposition of SrO and TiO₂, on homo-epitaxial and hetero-epitaxial growth of SrCuO₂ and YBa₂Cu₃O₇. A single-terminated surface turns out to be optimal with respect to a smooth morphology …