As the pulsatile cardiac blood flow drives the heart valve leaflets to open and close, the flow in the vicinity of the valve resembles a pulsed jet through a nonaxisymmetric orifice with a dynamically changing area. As a result, three-dimensional vortex rings with intricate topology emerge that interact with the complex cardiac anatomy and give rise to shear layers, regions of recirculation, and flow instabilities that could ultimately lead to transition to turbulence. Such complex flow patterns, which are inherently valve- and patient-specific, lead to mechanical forces at scales that can cause blood cell damage and thrombosis, increasing the likelihood of stroke, and can trigger the pathogenesis of various life-threatening valvular heart diseases. We summarize the current understanding of flow phenomena induced by heart valves, discuss their linkage with disease pathways, and emphasize the research advances required to translate in-depth understanding of valvular hemodynamics into effective patient ther...