Lithium niobate is a multi-functional material with wide reaching applications in acoustics, optics, and electronics. Commercial applications for lithium niobate require high crystalline quality currently limited to bulk and ion sliced material. Thin film lithium niobate is an attractive option for a variety of integrated devices, but the research effort has been stagnant due to poor material quality. Both lattice matched and mismatched lithium niobate are grown by molecular beam epitaxy (MBE) and studied to understand the role of substrate and temperature on nucleation conditions and material quality. Growth on sapphire produces partially coalesced columnar grains with atomically flat plateaus and no twin planes. A symmetric rocking curve shows a narrow linewidth with a full width at half-maximum (FWHM) of 8.6 arcsec (0.0024{deg}) which is comparable to the 5.8 arcsec rocking curve FWHM of the substrate, while the film asymmetric rocking curve is 510 arcsec FWHM. These values indicate that the individual grains are relatively free of long-range disorder detectable by x-ray diffraction (XRD) with minimal measurable tilt and twist and represents the highest structural quality epitaxial material grown on lattice mismatched sapphire without twin planes. Lithium niobate is also grown on lithium tantalate producing high quality coalesced material without twin planes and with a symmetric rocking curve of 193 arcsec, which is nearly equal to the substrate rocking curve of 194 arcsec. The surface morphology of lithium niobate on lithium tantalate is shown to be atomically flat by atomic force microscopy (AFM).