Lagrangian coherent structures (LCS) are best described as moving curves in a fluid that separate particles that have qualitatively different trajectories. For instance, particles that circulate in an ocean bay have a separate behavior from particles that go on by the bay and don't get caught up in the circulation. Interestingly, these two classes of particles are separated by a sharp, but moving curve. Similar structures are found in Hurricanes: which particle are going to get swept up in the Hurricane and which don't? Likewise in Jellyfish, some particles enter the underbelly of the jellyfish and bring nutrients, while others are swept downstream to help propel the jellyfish. The way blood flows over a clot, as revealed by LCS can indicate whether or not the clot is dangerous. This lecture will give examples of this sort, explain how the LCS are computed and are connected with other mathematical constructions, such as Smale horseshoes in dynamical systems.