Language impairment, or aphasia, is a disabling symptom that affects at least one third of individuals after stroke. Some affected individuals will spontaneously recover partial language function. However, despite a growing number of investigations, our understanding of how and why this recovery occurs is very limited. This Review proposes that existing hypotheses about language recovery after stroke can be conceptualized as specific examples of two fundamental principles. The first principle, degeneracy, dictates that different neural networks are able to adapt to perform similar cognitive functions, which would enable the brain to compensate for damage to any individual network. The second principle, variable neuro-displacement, dictates that there is spare capacity within or between neural networks, which, to save energy, is not used under standard levels of performance demand, but can be engaged under certain situations. These two principles are not mutually exclusive and might involve neural networks in both hemispheres. Most existing hypotheses are descriptive and lack a clear mechanistic account or concrete experimental evidence. Therefore, a better neurocomputational, mechanistic understanding of language recovery is required to inform research into new therapeutic interventions.