The complexity of the cells can be described and understood by a number of networks such as protein-protein interaction, cytoskeletal, organelle, signalling, gene transcription and metabolic networks. All these networks are highly dynamic producing continuous rearrangements in their links, hubs, network-skeleton and modules. Here we describe the adaptation of cellular networks after various forms of stress causing perturbations, congestions and network damage. Chronic stress decreases link-density, decouples or even quarantines modules, and induces an increased competition between network hubs and bridges. Extremely long or strong stress may induce a topological phase transition in the respective cellular networks, which switches the cell to a completely different mode of cellular function. We summarize our initial knowledge on network restoration after stress including the role of molecular chaperones in this process. Finally, we discuss the implications of stress-induced network rearrangements in diseases and ageing, and propose therapeutic approaches both to increase the robustness and help the repair of cellular networks.