A plug valve including a valve body defining a cavity and a pair of fluid passages intersecting the cavity, the cavity defining a tapered interior surface of the valve body, a plug extending within the cavity, the plug defining an exterior surface and a fluid passage adapted to be substantially aligned with the fluid passages of the valve body, and an insert extending within the cavity and circumferentially about at least a portion of the plug, the insert defining a pair of fluid passages substantially aligned with the fluid passages of the valve body, respectively. In several exemplary embodiments, the plug valve is used in oil or gas operations, such as, for example, the fracturing or gravel packing of a subterranean wellbore, with the plug valve being used to control the flow of fracturing and/or gravel-packing fluids. Exemplary embodiments of methods associated with the plug valve are also described.