We have investigated the microstructure of epitaxial YBa2Cu 3O7-x thin films on SrTiO3(100), LaGaO 3(100), and LaAlO3 (100) with particular emphasis on how the final microstructure is developed. Cross-sectional transmission electron microscopy as well as plan-view transmission electron microscopy combined with sputter depth profiling were used to study the change in microstructure with the increase in film thickness. For a thin film or near the substrate/film interface of a thick film, the YBa2Cu3O7-x film is composed primarily of grains oriented with c axis normal to the substrate surface and a small volume fraction of grains with c axis parallel to the substrate surface. As the film grows thicker, the c-axis parallel grains increase in size and grow over the top of the c-axis normal grains. The volume fraction of c-axis parallel grains increases rapidly as the film thickness increases and eventually the entire film surface is covered by c-axis parallel grains. However, the number density of the c-axis parallel grains remains constant throughout the whole film thickness. A growth model is proposed to explain the observed microstructure. Based on this model, a computer simulation is carried out. The simulated microstructure agrees well with the experimental result.