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In this chapter several aspects of the electronic and phonon structure are considered for the design and engineering of advanced thermoelectric materials. For a given compound, its thermoelectric figure of merit, zT, is fully exploited only when the free carrier density is optimized. Achieving higher zT beyond this requires the improvement in the material quality factor B. Using experimental data on lead chalcogenides as well as examples of other good thermoelectric materials, we demonstrate how the fundamental material parameters: effective mass, band anisotropy, deformation potential, and band degeneracy, among others, impact the thermoelectric properties and lead to desirable thermoelectric materials. As the quality factor B is introduced under the assumption of acoustic phonon (deformation potential) scattering, a brief discussion about carrier scattering mechanisms is also included. This simple model with the use of an effective deformation potential coefficient fits the experimental properties of real materials with complex structures and multi-valley Fermi surfaces remarkably well—which is fortunate as these are features likely found in advanced thermoelectric materials