We have grown aluminum nitride (AlN) on LSAT substrates for the first time using laser MBE and investigated its growth mechanism. It has turned out that surfaces of LSAT substrates become atomically flat by annealing in a UHV chamber. Reflection high energy electron diffraction observations and AFM measurements have shown that the growth of the AIN film starts with two-dimensional amorphous growth followed by three-dimensional epitaxial island growth. The epitaxial relationship between AlN and LSAT substrates is AlN(0 0 0 1)parallel to LSAT(1 1 1) and AlN[1 1 (2) over bar 0]parallel to LSAT[0 1 (1) over bar], which is rotated from the expected lattice matched alignment by 30 degrees along AlN[0 0 0 1]. X-ray diffraction measurements have revealed that the crystalline quality of the AlN film grown on LSAT is superior to that on a conventional Al2O3 substrate.