In this paper we set forth new exact analytical superluminal localized solutions to the wave equation for arbitrary frequencies and adjustable bandwidth. The formulation presented here is rather simple and its results can be expressed in terms of the ordinary, so-called "X-shaped waves". Moreover, by the present formalism we obtain the first analytical localized superluminal approximate solutions which represent pulses propagating in dispersive media. Our solutions may find application in different fields, like optics, microwaves, radio waves, and so on. © 2003 Elsevier B.V. All rights reserved.