We study two types of leaky wave antenna structures and the role of surface waves in their design. The first leaky wave antenna is a multi-dielectric structure of layered contrasting refractive indices. We review a rigorous derivation, by the present authors, of the radiation and surface wave powers excited by an embedded dipole source. It is shown that with this type of leaky wave antenna, the excitation of surface waves can drastically lower the antenna radiation efficiency. The latter is obtained in a closed form. The second leaky wave structure considered is based on the efficient excitation of surface waves on a dielectric slab. By introducing a suitable grating screen, the surface wave is smoothly converted into a leaky wave with a desirable radiation beam in the far-field. In this type of leaky wave antenna, efficient surface wave excitation is a key design step. We also review our perturbation analysis to find the radiation properties of planar periodic leaky wave antennas and certain design consideration are also recommended.