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Electron-phonon coupling affects the properties of two-dimensional (2D) materials significantly, leading to a series of novel phenomena. Inelastic light scattering provides a powerful experimental tool to explore electron-phonon interaction in 2D materials. This review gives an overview of the basic theory and experimental advances of electron-phonon coupling in 2D materials detected by Raman and Brillouin scattering, respectively. In the Raman scattering part, we review Raman spectroscopy studies of electron-phonon coupling in graphene, transition metal disulfide compounds, van der Waals heterostructures, strongly correlated systems, and 2D magnetic materials. In the Brillouin scattering part, we extensively introduce Brillouin spectroscopy in non-van der Waals 2D structures, including temperature sensors for phonons and magnons, interfacial Dzyaloshinsky-Moriya interaction and spin torque in multilayer magnetic structures, as well as exciton-polariton in semiconductor quantum well.