People with aphasia (PWA) present with language deficits including word retrieval difficulties after brain damage. Language learning is an essential life-long human capacity that may support treatment-induced language recovery after brain insult. This prospect has motivated a growing interest in the study of language learning in PWA during the last few decades. Here, we critically review the current literature on language learning ability in aphasia. The existing studies in this area indicate that (i) language learning can remain functional in some PWA, (ii) inter-individual variability in learning performance is large in PWA, (iii) language processing, short-term memory and lesion site are associated with learning ability, (iv) preliminary evidence suggests a relationship between learning ability and treatment outcomes in this population. Based on the reviewed evidence, we propose a potential account for the interplay between language and memory/learning systems to explain spared/impaired language learning and its relationship to language therapy in PWA. Finally, we indicate potential avenues for future research that may promote more cross-talk between cognitive neuroscience and aphasia rehabilitation.