Professor Lonseth was very interested in seeing that mathematics interact with other disciplines to improve the understanding of phenomena in these disciplines, and this lecture will be devoted to showing by three examples the importance of including the effects of capillarity in approximating multiphase flows in porous media. The first example involves a simple laboratory experiment and was responsible for reorienting an experimental procedure in a major petroleum research laboratory. The second relates to the mathematical description of multiphase flow in fractured media, where omitting capillarity leads to a seriously incorrect model. The third example, which concerns three-phase flow, exhibits a nonstandard shock that is incorrectly simulated (or not found) without capillarity. The presentation will not require expertise in simulating flows in porous media.