As one of the most important properties of glass-forming liquids, viscosity has drawn significant attention in both glass manufacturing and fundamental research. We review the recent scientific progress in viscosity of glass-forming systems, including both the liquid and glassy states. After the Vogel-Fulcher-Tammann (VFT) equation was introduced, many more efforts have been made to develop more accurate models to describe the temperature dependence of viscosity. In addition to the VFT equation, we also discuss three other viscosity models, viz., the Adam-Gibbs, Avramov-Milchev, and Mauro-Yue-Ellison-Gupta-Allan models. We compare the four viscosity models in terms of their theoretical underpinnings and ability to fit measured viscosity curves. The concept of fragility and the universality of the high-temperature viscosity limit are also discussed. Temperature-dependent constraint theory is introduced in detail as a powerful tool for predicting the composition dependence of viscosity. Some examples of the application of this approach to predict the glass transition temperature and fragility of various glass systems are shown. Topological constraint theory is not only of scientific interest, but also has important industrial applicability. We also discuss the thermal history dependence of viscosity in the glassy state. Some phenomenological models are briefly reviewed, while the main focus is given to the modified Mauro-Allan-Potuzak model, which can accurately predict the nonequilibrium viscosity as a function of temperature, thermal history, and composition. The correlation of viscosity with elasticity is described in terms of the shoving model. Some theoretical implications of the various viscosity models are discussed, including the concepts of the Kauzmann paradox and the ideal glass transition. Some of the evidence against the existence of these phenomena are discussed. We also review the link between glass relaxation and viscosity, that is, emphasizing that the viscosity equations presented in this review can also be used to model different types of relaxation effects based on the Maxwell relation.