Explore the mechanical modeling of living tissues in this seminar from the Society for Industrial and Applied Mathematics. Delve into multiphase models describing the dynamics of various cell types, liquids, and fibers in tissue repair and tumor growth. Examine the complexities of biological and mechanical effects in mathematical descriptions. Learn about the analysis of multiphase models based on Darcy's assumption, including compactness issues and the use of Aronson-Benilan estimates. Investigate the incompressible limit in special cases and associated free boundary problems. Gain insights from speaker Benoit Perthame on recent critical progress in this field, beneficial for mathematicians in relatively isolated places.