"Physical- and Mathematical-based reduced order modeling in computational hemodynamics" Alfonso Caiazzo The talk overviews few different model reduction approaches in blood flow modeling, a key issue to achieve realistic simulations in clinically relevant time. Three particular problems will be treated. First, the so-called lumped parameter models, used for the approximate simulation of the influence of secondary vessels on main arteries. Then, a mathematical modeling for blood flow through a porous interface will be described, an approach which can be used to simulate stents (tubular cardiovascular devices) and valves. Finally, we will discuss a geometrical based model order reduction for the simulation of pulmonary artery, used to investigate blood flow depending on different pressure conditions for a large set of patients.