Cognitive aging is characterized by a gradual decline in cognitive functioning. One of the most worrying deficits for older adults is a decreased capacity to memorize and remember new information. In this study, we assessed auditory short-term memory (STM), long-term memory (LTM), and working memory (WM) abilities of young and older adults using musical and numerical tasks. Additionally, we measured musical training and tested whether this capacity influences memory performance. Regarding STM, young adults scored higher than older adults when making same/different judgements of rhythmic sequences, but their performance was alike for melodic sequences. Higher levels of musical training were associated with enhanced STM capacity for melodic sequences. In relation to LTM, young adults outperformed older adults in identifying new musical sequences. Moreover, younger and older individuals with high musical training outperformed those with low musical training. No group differences were found in the recognition of previously memorized musical sequences. Finally, we found no group differences in WM capacity, although there was a non-significant tendency for young adults to outperform older adults. Overall, we found that aging differently affects several types of auditory memory and that, for certain musical memory tasks, a higher level of musical training provides significant advantages.