The behavior of leaky and surface modes in uniaxially anisotropic grounded slabs is investigated. First, a transverse magnetic and transverse electric modal parametric analysis of the structure is performed, based on dispersion relations, comparing the nondispersive and Drude/Lorentz dispersive anisotropic slabs with an isotropic nondispersive slab. This analysis reveals that in the case of the isotropic slab, the leaky wave pointing angle is restricted to the end-fire region. In contrast, it is shown, for the first time, that the proposed anisotropic dispersive grounded slab structure provides efficient (in particular highly directive) leaky wave radiation with a high design flexibility. Toward its lower frequencies, the dominant leaky mode provides full-space conical-beam scanning. At higher frequencies, it provides fixed-beam radiation (at a designable angle) with very low beam squint. A vertical dipole source is placed inside the slab to excite the relevant leaky wave mode. The radiation characteristics obtained for this structure confirms the novel low-beam squint and high-directivity operation of the dominant leaky mode. Further validation is included using the commercial software tool CST. The structure could be used to conceive antennas either for conical beam scanning (lower-frequency range) or for point-to-point communication and radar systems (higher-frequency range).