This paper presents the leaky wave phenomena associated with a waveguide structure that consists of a conductor-back dielectric slab covered with a superstrate made up of a two-dimensionally (2-D) electromagnetic band gap (EBG) structure. A guided wave has its energy bounced back and forth between the metallic ground plane and the EBG structure that is taken as a frequency-selective reflection mirror. Because of the finite thickness of the 2-D EBG superstrate, the guided wave will leak or radiate some of its energy into the air region above the waveguide structure to become a leaky wave. By the mode-matching method and the transverse resonance technique, the overall waveguide structure is formulated as a rigorous electromagnetic boundary value problem to yield an exact dispersion relation of the waveguide so that the complex propagation constant of a guided mode can be accurately determined, including the phase and attenuation constants. Additionally, the electric field distribution inside the waveguide and the far-field radiation pattern were also calculated to demonstrate the leaky wave phenomena of this waveguide from a microscopic point of view. On the basis of the excitation of leaky waves, the phenomenology concerning a class of directive antennas with EBG structure as superstrate was clarified in this research.