Patient-specific cardiovascular simulations can provide clinicians with predictive tools, fill current gaps in clinical imaging capabilities, and contribute to the fundamental understanding of disease progression. However, clinically relevant simulations must provide not only local hemodynamics, but also global physiologic response. This necessitates a dynamic coupling between the Navier–Stokes solver and reduced-order models of circulatory physiology, resulting in numerical stability and efficiency challenges. In this review, we discuss approaches to handling the coupled systems that arise from cardiovascular simulations, including recent algorithms that enable efficient large-scale simulations of the vascular system. We maintain particular focus on multiscale modeling algorithms for finite element simulations. Because these algorithms give rise to an ill-conditioned system of equations dominated by the coupled boundaries, we also discuss recent methods for solving the linear system of equations arising from these systems. We then review applications that illustrate the potential impact of these tools for clinical decision support in adult and pediatric cardiology. Finally, we offer an outlook on future directions in the field for both modeling and clinical application.