We have studied, using X-ray and transmission electron microscopy techniques, the texturing of YBa 2 Cu 3 07 superconducting thin films with c axis perpendicular to the lattice-mismatched ( 100) magnesium oxide substrate. The results were compared with epitaxial growth on the (100) SrTiO~ where the c axis of the film is either perpendicular or parallel to the substrate. Texturing with c axis perpendicular to the substrate occurs as a result of preferential grain growth in the "a" and "b" directions, whereas epitaxial growth involves lattice matching with the underlying substrate. Thin films of YBa2Cu~O7_ x were deposited using a pulsed-laser evaporation technique and were further annealed in oxygen to recover the superconducting properties, and to study the nature of textured growth. The grain growth in films was investigated as a function of annealing treatments. The high-temperature annealed films exhibited large textured grains (about 5-10/am) with c axis perpendicular to the substrate, but the (001) planes were found to be rotated randomly in the plane parallel to the substrate. The low-temperature annealed films showed small grains (-~ 100 nm) with no preferred texturing. From the microstructural variations between the high-and low-temperature annealed films, the process of grain growth in high-T, superconducting films was analysed. We propose a model based upon higher mobility of the a-b grain boundaries to explain the texturing of thin films with c axis perpendicular to the substrate.