The linear energy dispersion of graphene electrons leads to a strongly nonlinear electromagnetic response of this material. We develop a general quantum theory of the third-order nonlinear local dynamic conductivity of graphene , which describes its nonlinear response to a uniform electromagnetic field. The derived analytical formulas describe a large number of different nonlinear phenomena such as the third-harmonic generation, the four-wave mixing, the saturable absorption, the second-harmonic generation stimulated by a dc electric current, etc., which may be used in different terahertz and optoelectronic devices.