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Electroencephalography (EEG) has been widely used for studying brain function. As cortical signals recorded by the EEG are very weak, they are often obscured by motion artifacts and various non-brain physiological activities, such as eye movement, heart rhythms, and muscle activity, adversely affecting subsequent analysis and interpretation. Over the past decades, a number of techniques have been developed for preprocessing EEG recordings to improve the signal-to-noise ratio. However, based on our extensive literature survey, despite an increasing trend recently, only 14.72% of published studies on EEG preprocessing were involved with the removal of electromygraphic (EMG) artifacts. Given that ambulatory healthcare systems are continuously emerging, artifacts induced by muscle contraction become unavoidable, whereas in the past data tended to be collected in well-controlled clinical/laboratory settings. Motivated by the fact that EMG artifact removal is becoming an important issue to be addressed, we investigated the state-of-the-art muscle artifact removal methods systematically and comparatively, from the perspective of signal processing. In this review, we first present the signal characteristics from brain and muscle activity, and highlight the importance of this issue for subsequent artifact removal. We then provide an overview and taxonomy of representative methods. Based on the results in reported studies, we describe the pros and cons of different methods and give suggestions on selecting a suitable technique in different scenarios (e.g., single-channel, few-channel, and multichannel). Finally, we discuss remaining challenges and provide feasible recommendations for further exploration in this field.