Modeling of fluid flows through porous deformable media is relevant for many applications in biology, medicine and bioengineering, including blood flow through human tissues and fluid flow inside cartilages and bones. These fluid-structure mixtures are described mathematically by nonlinear poro-visco-elastic systems in bounded domains, with mixed boundary conditions. In this talk, I will present well-posedness results for these models, and investigate the influence of structural viscoelasticity on the solution and on the regularity requirements for the forces present in the system. Lastly, I will discuss applications of our results in the case of blood flow through ocular tissues, and the connections to the development of glaucoma.