Animal movement patterns have long been the subject of mathematical and ecological interest. How do individual behavioral decision rules translate into macroscale patterns of space use such as foraging, patrolling or territories? I will show how mechanistic models, using random walks, stochastic processes, first passage time analysis and partial differential equations can be used to connect underlying processes to the observed patterns. Here interactions are complex and may involve memory of past events, as well as a cognitive map. I will make applications to a spectrum of different emerging patterns, ranging from territories in Amazonian birds to patrolling in wolves.