Patient-specific models of blood flow in the coronary arteries have entered clinical practice worldwide to aid in the diagnosis and management of heart disease. This technology leverages modern AI-based image segmentation methods to extract the geometry of the coronary arteries from noninvasive computed tomography volumetric imaging, computational physiology methods to define boundary conditions and computational fluid dynamic techniques to compute coronary artery flow and pressure. The methods described herein have been used to compute the fractional flow reserve (FFR) in the coronary arteries, namely the ratio of pressure in a coronary artery to the reference aortic pressure under conditions of maximum hyperemia induced by the intravenous administration of a vasodilator to mimic increased coronary artery blood flow as occurs during physical activity. The fractional flow reserve derived from coronary computed tomography angiography (CCTA) using patient-specific models of coronary artery blood flow is referred to as FFRCT and has been validated against invasive measurements using pressure wires in several hundred patients and subsequently evaluated in clinical trials in several thousand patients. Patient-specific models of coronary artery blood flow can be used to predict outcomes of coronary artery revascularization procedures, to identify patients at risk of having a heart attack, and to model tissue perfusion of the microcirculation by coupling with emerging models of cardiac function. This paper provides a state-of-the-art review of computational methods for, and applications of, quantifying blood flow in patient-specific models of the coronary arteries.