Remarkably, an astrophysical black hole has only two attributes: its mass and its spin angular momentum. Spin is often associated with the exotic behavior that black holes manifest such as the production of relativistic and energetic jets. In this thesis, we advance one of the two primary methods of measuring black hole spin, namely, the continuum-fitting method by (1) improving the methodology; (2) testing two foundational assumptions; and (3) measuring the spins of two stellar-mass black holes in X-ray binary systems. Methodology: We present an empirical model of Comptonization that self-consistently generates a hard power-law component by upscattering thermal accretion disk photons as they traverse a hot corona. We show that this model enables reliable measurements of spin for far more X-ray spectral data and for more sources than previously thought possible. Testing the foundations: First, by an exhaustive study of the X-ray spectra of LMC X–3, we show that the inner radius of its accretion disk is constant over decades and unaffected by source variability. Identifying this fixed inner radius with the radius of the innermost stable circular orbit in general relativity, our findings establish a firm foundation for the measurement of black hole spin. Secondly, we test the customary assumption that the inclination angles of the black-hole’s spin axis and the binary’s orbital axis are the same; for XTE J1550–564 we show that they are aligned to within \(12^{\circ}\) by modeling the kinematics of the large-scale jets of this microquasar. Measuring spins: We have made the first accurate continuum-fitting spin measurements of the black hole primaries in H1743–322 and XTE J1550–564. For this latter black hole, we have also measured its spin using the other leading method, namely, modeling the broad red wing of the \(Fe K\alpha\) line. As we show, these two independent measurements of spin are in agreement. / Astronomy