From bird flocks to fish schools, animal groups exhibit a remarkable ability to manage challenging tasks that individuals could not manage on their own.  Despite limitations on individual-level sensing, computation, and actuation, and with no centralized instruction, animal groups make decisions quickly, accurately, robustly and adaptively in an uncertain, changing environment.  I will describe recent analytically tractable models and methods for studying the mechanisms of collective migration and collective decision-making in animal groups.  A focus is on bifurcation analyses, which systematically elucidate the dependence of performance of the collective dynamics on parameters that model the system and the environment.