The invariance problem refers to the challenge that listeners face when confronted with acoustic variability in speech sounds as they attempt to map these sounds to few phonological categories. This problem is compounded in the adult non-native language learner, in that the listener must overcome processing biases that prevent the perception of key differences among tokens within the native language category. Evidence from behavioral and neural data suggests that phonetic category learning can be described as a two-stage process, in which categorical sensitivity first emerges in frontal brain structures, and subsequently "tunes" perceptual areas in the temporal lobes. Moreover, sleep appears to modulate non-native category learning. Taken together, these findings provide new insight regarding the overall trajectory of phonetic learning that results in heightened sensitivity to non-native sounds.