The cocktail party problem, i.e., tracing and recognizing the speech of a specific speaker when multiple speakers talk simultaneously, is one of the critical problems yet to be solved to enable the wide application of automatic speech recognition (ASR) systems. In this overview paper, we review the techniques proposed in the last two decades in attacking this problem. We focus our discussions on the speech separation problem given its central role in the cocktail party environment, and describe the conventional single-channel techniques such as computational auditory scene analysis (CASA), non-negative matrix factorization (NMF) and generative models, the conventional multi-channel techniques such as beamforming and multi-channel blind source separation, and the newly developed deep leaing-based techniques, such as deep clustering (DPCL), the deep attractor network (DANet), and permutation invariant training (PIT). We also present techniques developed to improve ASR accuracy and speaker identification in the cocktail party environment. We argue effectively exploiting information in the microphone array, the acoustic training set, and the language itself using a more powerful model. Better optimization objective and techniques will be the approach to solving the cocktail party problem.