Modelling the evolution of biological networks is a major challenge. Biological networks are usually represented as graphs; evolutionary events include addition and removal of vertices and edges, but also duplication of vertices and their associated edges. Since duplication is viewed as a primary driver of genomic evolution, recent work has focused on duplication-based models. Missing from these models is any embodiment of modularity, a widely accepted attribute of biological networks. Some models spontaneously generate modular structures, but none is known to maintain and evolve them. We describe NEMo (Network Evolution with Modularity), a new model that embodies modularity. NEMo allows modules to emerge and vanish, to fission and merge, all driven by the underlying edge-level events using a duplication-based process. We introduce measures to compare biological networks in terms of their modular structure and use them to compare NEMo and existing duplication-based models and to compare both generated and published networks.