Topology is emerging as an important new tool for understanding our world. Computational homology, for example, has become standard for analyzing the connectivity of large-dimensional data sets. Here I present another approach, which is more dynamical in nature. The trajectories of 'particles,' whether oceanic floats or people, can be regarded as mathematical objects called braids. By using traditional concepts from topological dynamics, such as topological entropy, we gain insight into the inherent complexity of motion.