We claim that networks are created according to the priority attachment mechanism. We introduce a simple model, which uses the priority attachment to generate both synthetic and close to empirical networks. Priority attachment is a mechanism, which generalizes previously proposed mechanisms, such as small world creation or preferential attachment, but we also observe its presence in a range of real-world networks. In this paper, we show that by using priority attachment we can generate networks of very diverse topologies, as well as re-create empirical ones. An additional advantage of the priority attachment mechanism is an easy interpretation of the latent processes of network formation. We substantiate our claims by performing numerical experiments on both synthetic and empirical networks. The two main contributions of the paper are: the development of the priority attachment mechanism, and the design of Priority Rank: a simple network generative model based on the priority attachment mechanism.