This paper overviews a series of recent advances in adaptive processing and learning for audio source separation. In real world, speech and audio signal mixtures are observed in reverberant environments. Sources are usually more than mixtures. The mixing condition is occasionally changed due to the moving sources or when the sources are changed or abruptly present or absent. In this survey article, we investigate different issues in audio source separation including overdetermined/underdetermined problems, permutation alignment, convolutive mixtures, contrast functions, nonstationary conditions and system robustness. We provide a systematic and comprehensive view for these issues and address new approaches to overdetermined/underdetermined convolutive separation, sparse learning, nonnegative matrix factorization, information-theoretic learning, online learning and Bayesian approaches.