We claim that networks are created according to the priority attachment mechanism and we show 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 recreate empirical networks. 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 synthetic and empirical networks. The two main contributions of the paper are: the introduction of the priority attachment mechanism, and the design of the Priority Rank: a simple network generative model based on the priority attachment mechanism.