This thesis describes procedures and theory for a family of one-sided ultrasonic methods for determining third order elastic constants (TOEC) using sets of angle-beam wedges mounted on one side of a specimen. The methods are based on the well-known acoustoelastic effect, which is the change of wave speed with applied loads and is a consequence of the mechanical nonlinearity of a material. Increases in material nonlinearity have been correlated to the progression of damage, indicating that tracking changes in TOECs may provide a practical means of monitoring damage accumulation at the microstructural level prior to formation of macroscopic defects. Ultrasonic methods are one of the only ways to measure TOECs, and most prior techniques have utilized wave propagation paths parallel and perpendicular to the loading directions. A few additional ultrasonic techniques reported in the literature have employed oblique paths but with immersion coupling. These reported techniques are generally unsuitable for field implementation. The one-sided contact approach described here is applicable for in situ measurements of TOECs and thus lays the foundation for tracking of TOECs with damage. Theory is reviewed and further developed for calculating predicted velocity changes, and thus time shifts, as a function of uniaxial tensile loading for longitudinal, shear vertical, and shear horizontal waves in the context of angle-beam transducers mounted on the surface of the specimen. A comparison is made to published results where possible. The inverse problem of determining the three TOECs of an isotropic material from three measurements employing three different angle beam configurations is comprehensively analyzed. Four configurations providing well-posed solutions are identified and examined. A detailed sensitivity analysis is carried out to identify the best mounting configuration, wave mode combinations, refracted angles and geometry requirements for recovering the three TOECs. Two transducer mounting configurations are considered: (1) attached (glued-on) transducers potentially suitable for in situ monitoring, and (2) floating (oil-coupled) transducers potentially suitable for single measurements. Limited experimental results are presented for the attached case using two longitudinal measurements and one shear vertical measurement. The floating case experiments utilized three of the four well-posed solutions, and measurements were made on several aluminum alloys and low carbon steel. Key experimental issues are identified and discussed for both transducer mounting configurations.