During the neutron detection process, owing to the effects of inelastic scattering and slow neutron capture, a neutron-gamma mixed radiation field is formed, which increases the complexity of neutron detection. Organic scintillators are widely used in neutron detection because of their high flashing efficiency, short decay time, and high detection efficiency. Pulse shape discrimination (PSD) is a key technology for discriminating neutrons and gamma rays according to the difference in pulse shape caused by the difference in particle decay time in organic scintillators. Traditional PSD methods include time-domain and frequency-domain discrimination methods. In recent years, various machine-learning techniques applied to neutron-gamma discrimination have achieved better results. To better use organic scintillators and the corresponding neutron-gamma discrimination methods in neutron detection, we conducted a comprehensive analysis of the glowing mechanism of organic scintillators, PSD principle, organic scintillator types, and neutron-gamma discrimination methods and investigated the performance evaluation indexes of organic scintillators and neutron-gamma discrimination methods. Finally, the future development directions of organic scintillators and neutron-gamma discrimination methods were examined.