21-cm cosmology concerns itself with the last, unmeasured epochs of the Universe, between the decoupling of the CMB and the end of Reionization in particular. These epochs include the cosmic Dark Ages, before any luminous astrophysical structures had formed, Cosmic Dawn, when the first sources and galaxies appeared, and the Epoch of Reionziation, when they reionized the intergalactic medium. By measuring the hyperfine, highly redshifted 21-cm line of neutral Hydrogen during these epochs, it is possible to map the thermal and ionization state of the entire Universe, describe the first sources, map structure formation, and even test dark matter and cosmology models in unique ways and epochs, such as during the Dark Ages. Formidable systematics, however, must be removed before an unambiguous detection is made. In this thesis we present novel models, statistical tests, and analyses for two of the primary systematics, including the galactic foreground and the radio antenna beam, as well as methods for robustly constraining and removing them from radio spectrometer data aimed at detecting the 21-cm signal. We also present an analysis describing how warm dark matter and population III stars affect the shape of the sky-averaged 21-cm signal and an analysis of their detectability in the presence of degeneracies with other astrophysical model parameters and systematic noise. Lastly, we present new results from the first radio telescope of NASA’s to land on the Moon, ROLSES-1, including potential techno-signatures detected from the Earth and constraints upon the isotropic galactic background spectrum, both the first measurements of their kind from the surface of the Moon. We present the methods for calibrating, cleaning, and reducing the raw data, as well as the statistical methods used to extract antenna parameter values and the amplitude of the galactic spectrum. Lastly, we outline some methods for use in modeling and 21-cm signal extraction in the future, and how our methods may be applied to upcoming lunar radio telescopes.