Cognitive aging is characterized by the gradual decline of a number of abilities, such as attention, executive functioning, and memory. Research on memory aging has reported age-related deficits in short-term (STM), long-term (LTM), and working memory (WM) and linked these to structural and functional changes in the brain that occur with aging. However, only a few studies have drawn direct comparisons between these memory subsystems in the auditory domain. In this study, we assessed auditory STM, LTM, and WM abilities of young (under 25 years of age) and older (over 60 years of age) adults using musical and numerical tasks. In addition, we measured musical training history and tested its modulating effects on auditory memory performance. Overall, we found that older adults underperformed in specific memory tasks, such as STM related to discrimination of rhythmic sequences, LTM associated with identification of novel musical sequences, and numerical WM. Furthermore, we observed a positive influence of musical training on certain memory tasks involving music. In conclusion, aging differentially affects several types of auditory memory, and in the case of specific musical memory tasks, a higher level of musical training provides significant advantages.