Valvular heart disease is a serious medical condition that requires accurate diagnosis and effective treatment. Fluid-Structure Interaction (FSI) is crucial to understanding heart valve hemodynamics and biomechanics. Achieving predictivity, robustness, and modeling flexibility in simulating real-world FSI problems is essential to developing effective treatments for valvular heart disease. However, this involves unique challenges, including realistic geometric models, patient customization, stability and transition of blood flows, nonlinearity in large-deformation structures, and conservation properties at the coupling interface. To address these challenges, a high-fidelity immersogeometric FSI framework has been developed. This framework seamlessly integrates complex parametric design with multiphysics simulations and provides a sustainable, open-source paradigm for fully automated computational analysis. The FSI framework has enabled studies in cardiovascular applications, such as investigating the significance of arterial wall deformation in vascular flow, understanding heart valve leaflet flutter in thin tissues, studying transcatheter valve hemodynamics and biomechanical functionality, and quantifying the impact of geometric characteristics such as leaflet curvature and free edge length on the effective orifice and coaptation areas. Research findings will be summarized, and future opportunities will be presented.