Abstract: The problem of the epitaxial growth of the high temperature superconducting R1+xBa2xCu3O7δ (R = Y or rare earth except Ce and Tb) films has been addressed. Using in situ ultra high vacuum Scanning Tunneling Microscopy (UHV-STM) we have studied the role of cationic substitution and substrate mismatch on the growth mode of stoichiometric and Nd-rich Nd1+xBa2xCu3O7δ thin films. The results are compared to the growth of Y1Ba2Cu3O7δ, Dy1Ba2Cu3O7δ and Gd1Ba2Cu3O7δ epitaxial films. Two main phenomena are investigated: a) the first stage of the direct nucleation on the substrate and b) the crossover between 2D and 3D growth upon increasing the film thickness. At the first stage of the growth, pseudo-cubic perovskite (Re,Ba)CuO3 nuclei are formed. While they disappear after the growth of a few nm in stoichiometric films, they persist on the surface of Nd-rich films of up to 110 nm thickness. Stoichiometric R1+xBa2xCu3O7δ films exhibit a rough morphology with increasing thickness due to island growth mode, whereas Nd-rich films remain smooth and continue to grow layer by layer. It is proposed that linear defects (like anti-phase boundaries), which are formed due to the misalignment of growth fronts, are the source of screw dislocations in stoichiometric films. In Nd-rich films, linear defects are eliminated through the insertion of (Nd,Ba)CuO3 extra layers without introduction of any screw dislocations.