Aluminum nitride (AlN) is a direct bandgap semiconductor with a ultrawide bandgap width (i.e., 6.2 eV), a high thermal conductivity (i.e., 3.4 W/(cm·K)), a high surface acoustic rate (i.e., VL=10.97×105 cm/s, VT=6.2×105 cm/s), a high breakdown field strength as well as stable physical and chemical properties. AIN is an ideal substrate for ultraviolet/deep ultraviolet (DUV) luminescent materials. AlxGa1-xN materials made from AlN can achieve a continuous luminescence in a wavelength range of 200-365 nm. AlN crystal as one of developed semiconductor materials with a great potential can be used for high voltage, high temperature and high frequency electronic devices. This review introduced the heteroepitaxial growth principle of AlN single crystal by a physical vapor transport (PVT) method. Recent research progress on the heteroepitaxial growth of AlN on silicon carbide (SiC) substrate was represented based on the corresponding studies on the AlN growth on SiC substrates and the growth of AlN crystals on AlN/SiC substrates and off-orientation SiC substrates. Some challenges and opportunities of growing AlN single crystal on SiC substrate were briefly described, and the future development of AlN materials was prospected as well.