We examine early olfactory processing in the vertebrate and insect olfactory systems, using a computational perspective. What transformations occur between the first and second olfactory processing stages? What are the causes and consequences of these transformations? To answer these questions, we focus on the functions of olfactory circuit structure and on the role of time in odor-evoked integrative processes. We argue that early olfactory relays are active and dynamical networks, whose actions change the format of odor-related information in very specific ways, so as to refine stimulus identification. Finally, we introduce a new theoretical framework ("winnerless competition") for the interpretation of these data.