In this work, using the compact density matrix approach, we study the linear and nonlinear magneto-optical properties of monolayer molybdenum disulfide (MoS2) via an investigation of the absorption coefficients (MOACs) and refractive index changes (RICs). The results are presented as functions of photon energy and external magnetic field. Our results show that the MOACs and the RICs appear as a series of peaks in the inter-band transitions between Landau levels, while the intra-band transitions result in only one peak. Because of the strong spin-orbit coupling, the peaks caused by the spin-up and -down states are different. With the increase in the magnetic field, both MOACs and RICs give a blue-shift and reduce in their amplitudes. These results suggest a potential application of monolayer MoS2 in the optoelectronic technology, magneto-optical, valleytronic, and spintronic devices.