Modeling real-world networks is a key focus area of Network Science. I will first provide a brief overview into the state-of-the-art of network modeling, then turn to the discussion of network evolution models. In current network-evolution models, the degree of each node varies or grows arbitrarily, yet there are many networks for which a different description is required. In some networks, node degree saturates, such as the number of active contacts of a person, and in some it is fixed, such as the valence of an atom in a molecule. I will introduce a novel family of network growth processes that preserve node degree (DPG), resulting in structures substantially different from those reported previously. We demonstrated that, despite it being an NP (non-deterministic polynomial time)-hard problem in general, the exact structure of most real-world networks can be generated from degree-preserving growth. We also show that this process can create scale-free networks with arbitrary exponents, however, without preferential attachment. If preferential attachment is an effective rich-gets-richer mechanism applied to connectivity formation, then degree-preserving growth is a type of “tinkering” mechanism, a property observed in many real systems, e.g., by the Nobel Laurate F. Jacob in: “Evolution and Tinkering”, Science, 196, 1161 (1977)). Finally, I will present applications of DPG to epidemics control via network immunization, viral marketing, knowledge dissemination, the design of molecular isomers with desired properties and to a problem in number theory.