Congenital amusia is a neuro-developmental disorder of music perception and production, with the observed deficits contrasting with the sophisticated music processing reported for the general population. Musical deficits within amusia have been hypothesized to arise from altered pitch processing, with impairments in pitch discrimination and, notably, short-term memory. We here review research investigating its behavioral and neural correlates, in particular the impairments at encoding, retention, and recollection of pitch information, as well as how these impairments extend to the processing of pitch cues in speech and emotion. The impairments have been related to altered brain responses in a distributed fronto-temporal network, which can be observed also at rest. Neuroimaging studies revealed changes in connectivity patterns within this network and beyond, shedding light on the brain dynamics underlying auditory cognition. Interestingly, some studies revealed spared implicit pitch processing in congenital amusia, showing the power of implicit cognition in the music domain. Building on these findings, together with audiovisual integration and other beneficial mechanisms, we outline perspectives for training and rehabilitation and the future directions of this research domain.