With increasing radius of critical nuclei at low supersaturations the substrate parameters misfit and misorientation become crucial for the initial growth stage and the growth mode of thin films. The importance of substrate misfit and misorientation control in liquid phase epitaxy (LPE) and physical vapor deposition (PVD) of rare earth cuprates is discussed, and critical supersaturations to start growth on mismatched substrates are estimated. Good agreement with experiment was found for the a/c-transition of YBa 2Cu 3O 7− δ (YBCO) on (1 1 0)NdGaO 3 in LPE at a critical undercooling (supersaturation) Δ T≈3 K. For tunnel device technology, atomically flat surfaces with interstep distances (monosteps) of about 10 μm are needed. This requires a substrate misfit ⩽0.08%, a relative supersaturation ⩽0.18%, and a substrate misorientation ⩽0.02° to avoid step bunching. For c-oriented YBCO films on (1 1 0)NdGaO 3, with a misfit of 0.28%, monospirals with a maximum interstep distance of y 0=2.5 μm between monosteps can be grown.