Obstructive sleep apnea (OSA) is a sleep disorder characterized by periodic episodes of partial or complete upper airway obstruction caused by narrowing or collapse of the pharyngeal airway despite ongoing breathing efforts during sleep. Fall in the blood oxygen saturation and cortical arousals are prompted by this reduction in the airflow which lasts for at least 10 seconds. Impaired labor performance, debilitated quality of life, excessive daytime sleepiness, high snoring, and tiredness even after a whole night’s sleep are the primary symptoms of OSA. In due course, the long-standing contributions of OSA culminate in hypertension, arrhythmia, cerebrovascular disease, and heart failure. The traditional diagnostic approach of OSA is the laboratory-based polysomnography (PSG) overnight sleep study, which is a tedious and labor-intensive process that exaggerates the discomfort to the patient. With the advent of computer-aided diagnosis (CAD), automatic detection of OSA has gained increasing interest among researchers in the area of sleep disorders as it influences both diagnostic and therapeutic decisions. The research literature on sleep apnea published during the last decade has been surveyed, focusing on the varied screening approaches accustomed to identifying OSA events and the developmental knowledge offered by multiple contributors from the software perspective. The current study presents an overview of the pathophysiology of OSA, the detection methods, physiological signals related to OSA, the different preprocessing, feature extraction, feature selection, and classification techniques employed for the detection and classification of OSA. Consequently, the research challenges and research gaps in the diagnosis of OSA are identified, critically analyzed, and presented in the best possible light.