CUORE---the Cryogenic Underground Observatory for Rare Events---is an experiment searching for the neutrinoless double-beta ($0\nu\beta\beta$) decay of $^{130}$Te, based at the Laboratori Nazionali del Gran Sasso in Italy. The detector consists of 988 5$\times$5$\times$5 cm$^3$ TeO$_2$ crystals operated as bolometers at temperatures of $\sim$10 mK inside the world's largest and most powerful dilution refrigerator. CUORE began physics data collection in the spring of 2017, and has recently released its first limit on the $0\nu\beta\beta$ decay half-life of $^{130}$Te from 24 kg $\cdot$ y isotope exposure ($\sim$2 months live time). This result---T$^{0\nu}_{1/2} > 1.5 \cdot 10^{25}$ y (Bayesian) and T$^{0\nu}_{1/2} > 2.3 \cdot 10^{25}$ y (Frequentist) at $90 \%$ C.L.---is the most stringent to date and, together with two alternative analyses necessary to calculate it, forms the centerpiece of this thesis. In the future, with five years of live time, CUORE is projected to reach a median sensitivity of $9 \cdot 10^{25}$ y on this half-life. Besides the main physics conclusions, in this work I present an analysis modeling the spectral line shape of the bolometer, which is used for constructing the region of interest fit PDF. Additionally, I discuss the major CUORE hardware projects to which I have contributed in a significant way. Specifically, these are our world-leading cryostat, a cryogenic feedback temperature control system, a radon-free detector installation environment, and a room temperature detector calibration system.