We report a study of a novel quantum well structure, the random-period superlattice quantum well (RSLQW), consisting of an AlAs/GaAs superlattice (SL) sandwiched between two AlGaAs layers, where the periods in the SL are intentionally randomized. Photoluminescence (PL) measurements are performed to probe the optical properties of the RSLQW structures. Theoretical calculations are employed to correlate the type of transition and transition energy with the experimental observations. Phonon sidebands are identified in the PL spectra of RSLQWs, in which type II transitions are predicted by calculation results. Compared to the findings in periodic superlattice quantum wells, some RSLQWs show significantly stronger PL signals. However, all the RSLQW samples exhibit larger degradation of radiative recombination efficiency with increasing temperature.