We have developed a new family of limited diffraction electromagnetic X-shaped waves based on the scalar X-shaped waves discovered previously. These waves are diffraction-free in theory and particle-like (wave packets), in that they maintain their shape as they propagate to an infinite distance. The “X waves” possess (theoretically) infinitely extended “arms” and —at least, the ones studied in this paper— have an infinite total energy: therefore, they are not physically realizable. However, they can be truncated in both space and time and “approximated” by means of a finite aperture radiator so to get a large enough depth of interest (depth of field). In addition to the Maxwell equations, X wave solutions to the free Schroedinger equation are also obtained. Possible applications of these new waves are discussed. Finally, we discuss the appearance of the X-shaped solutions from the purely geometric point of view of the special relativity theory.