Most electronic materials exhibit a single type of conduction polarity, either holes (p-type) or electrons (n-type), along their principle axes. In the introductory chapter, we establish an overview of the chemical design principles for creating materials that can simultaneously exhibit n-type conduction and p-type conduction along different directions of the single crystal. Materials that exhibit such behavior have the potential to impact a broad range of electronic and energy-harvesting technologies. We show using first-principles predictions that the AMX (A = Ca, Sr, Ba; M = Cu, Ag, Au; X = P, As, Sb) compounds consisting of MX honeycomb layers separated by A cations can exhibit p-type conduction in-plane and n-type conduction cross-plane via either mechanism depending on the doping level. In Chapter 2, we uncover a new material genus with this behavior, now originating from the Fermi surface topology of a single band. NaSn2As2, a layered metal, has a Fermi surface with this topology. This direction-dependent carrier polarity, which we label goniopolarity, and this Fermi surface topology, are expected in many metals and semiconductors whose electronic structure is at the boundary between two and three dimensions. In Chapter 3, we show that Re4Si7, which has the largest axis-dependent conduction anisotropy to date, can be implemented in transverse thermoelectric devices. Thermoelectric devices directly convert heat to electrical power using a p-type and n-type semiconductor connected thermally in parallel and electrically in series and represent a green energy source for waste heat recovery. The principal challenges in current thermoelectric power generation modules is the availability of stable, diffusion-resistant, lossless electrical and thermal metal-semiconductor contacts that do not degrade at the hot end nor cause reductions in device efficiency. Transverse thermoelectric devices, in which a thermal gradient in a single material induces a perpendicular voltage, promise to overcome these problems. However, the measured material transverse thermoelectric efficiencies, zxyT, of nearly all materials to date has been far too low to confirm these advantages in an actual device. Here, we show that single crystals of Re4Si7, an air-stable, thermally robust, layered compound, have a transverse zxyT of 0.7±0.15 at 980 K, a value that is on par with existing commercial longitudinal thermoelectrics today. Through constructing and characterizing a transverse power generation module, we prove that extrinsic losses through contact resistances are minimized in this geometry, and that no electrical contacts are needed at the hot side. Finally, in Chapter 4 we further investigate the performance of Re4Si7 transverse power generators using different device geometries. The effect of two different sample geometries, quasi-adiabatic and quasi-isothermal, was evaluated based on the performance of transverse thermoelectric power generation using samples of Re4Si7. Transverse thermoelectric conversion efficiency measurements made at several different temperatures showed a distinct difference in transverse figure of merit, with the quasi-isothermal geometry showing a greater thermal efficiency and transverse figure of merit than the quasi-adiabatic geometry, in agreement with mathematical predictions for ZxyT in these two device configurations. This work will help pave the way for future device fabrication of efficient transverse thermoelectric power generation modules using single crystals with axis-dependent conduction, and highlights the importance of device geometry for maximizing the conversion of heat to electricity.