AlN/GaN multiple quantum wells (MQWs) with a well thickness of 26 Å have been grown by metal-organic chemical-vapor deposition. A specially designed photoluminescence (PL) spectroscopy system, which is capable of measuring picosecond time-resolved PL up to 6.2 eV, has been employed to probe the optical properties as well as the carrier transfer and decay dynamics in these MQWs. Optical transitions at 4.039 and 5.371 eV at T=10 K, resulting from the interband recombination between the electrons and holes in the n=1 and n=2 subbands in the wells, have been observed. The band-offset parameter for the AlN/GaN heterostructure has been obtained by comparing the experimental results with the calculations. Carrier dynamics including the relaxation of the electrons and holes from the n=2 and n=1 subband in the conduction and valence bands and the decay lifetimes of the interband transitions have also been measured and analyzed. Detailed subband structures for both the conduction and valence bands in the wells were determined. The implications of our findings on the potential applications of AlN/GaN quantum wells have been discussed.