Understanding the origins of complexity is a key challenge in many sciences. Although networks are known to underlie most systems, showing how they contribute to well-known phenomena remains an issue. Here, we show that recurrent phase transitions in network connectivity underlie emergent phenomena in many systems. We identify properties that are typical of systems in different connectivity phases, as well as characteristics commonly associated with the phase transitions. We synthesize these common features into a common framework, which we term dual-phase evolution (DPE). Using this framework, we review the literature from several disciplines to show that recurrent connectivity phase transitions underlie the complex properties of many biological, physical and human systems. We argue that the DPE framework helps to explain many complex phenomena, including perpetual novelty, modularity, scale-free networks and criticality. Our review concludes with a discussion of the way DPE relates to other frameworks, in particular, self-organized criticality and the adaptive cycle.