This talk focuses on differential problems describing the flow of a viscous fluid in deformable domains. Such problems include flow in compliant tubes, often adopted for the modeling of arterial blood flow, and flow through deformable porous media, often adopted for the modeling of tissue perfusion. The mixed hyperbolic-parabolic-elliptic nature of these systems guides the study of their well-posedness and the design of efficient numerical methods for their approximate solution. Theoretical and numerical results will be presented, along with their impact on real-world applications. In addition, the need of capturing the simultaneous effects of local and nonlocal factors in human cardiovascular physiology requires the coupling between systems of partial differential equations (PDEs), such as those described above, with systems of ordinary differential equations (ODEs) describing the blood flow in the systemic circulation. Open questions and recent results concerning the PDE/ODE coupling in fluid flow applications will be discussed from the theoretical and computational viewpoints.