Electroencephalography (EEG) is one of the most important biomedical signals in the evaluation of brain activities in frontier research and clinical practice. Based on EEG processing, electrical activity from human brain could be decoded and understood, which benefits many scenarios such as seizure detection, sleep staging and motor imagery. In this review, Application Specific Integrated Circuits (ASICs)  for EEG processing in literature of the past decades are carefully investigated. The EEG processors are divided into four types according to their architectures. A thorough analysis is performed on all the four EEG processor types, including quantitative cross-task comparisons in terms of energy efficiency,  accuracy, and hardware complexity. The challenges and opportunities for the researches in hardware EEG  processing are also included.