There are a number of interesting and important biological processes that are best modelled as two-phase material mixtures. These include mucin exocytosis and transport, blood clot formation and biofilm formation. These all involve the interplay between flow, physical structure, mechanics and chemistry in a environment with complex dynamic geometry. The mathematical description of these processes requires equations describing multiphase flow, the evolution of composition and structure, and the relationship between stresses and composition/ structure (i.e., constitutive relations). Additionally, these equations of motion must properly account for interactions of the complex materials with dynamic physical boundaries, moving interfaces between materials with markedly different physical properties, and typically include strongly nonlinear constitutive relations or rate expressions. In this talk, I will describe two features of mucus: the dynamics of mucus vesicle exocytosis and its transport of acid against an acid gradient. The short story is as follows: Mucin is packaged into vesicles at very high concentration (volume fraction) and when the vesicle is released to the extracellular environment, the mucin expands in volume by two orders of magnitude in a matter of seconds. This rapid expansion is mediated by the rapid exchange of calcium and sodium that changes the crosslinking structure of the mucin polymers, thereby causing it to swell. I will describe a model of gel swelling and deswelling that accounts for these features, and is an interesting free boundary problem. One of the important functions of the mucus lining of the stomach is to allow digestion of food to take place without the lining of the stomach being digested. An intriguing question is how acid can be released into the lumen of the stomach while maintaining a low concentration of hydrogen ions near the epithelial lining. A possible answer is that the flow of acid against its gradient is mediated by buffering by mucin. When mucin is secreted it rapidly binds hydrogen, but when it reaches the lumen where the pH is low, mucin is degraded by pH-activated pepsin, releasing its acid. The model associated with this process includes a free boundary problem to determine the thickness of the mucus layer and its acid-protective ability.