Nature and the world of human technology are full of networks. People like to draw diagrams of networks: flow charts, electrical circuit diagrams, signal-flow graphs, Bayesian networks, Feynman diagrams and the like. Mathematically minded people know that in principle these diagrams fit into a common framework: category theory. But we are still far from a unified theory of networks. After an overview, we will look at three portions of the jigsaw puzzle in three separate talks: I. Electrical circuits and signal-flow graphs. II. Stochastic Petri nets, chemical reaction networks and Feynman diagrams. III. Bayesian networks, information and entropy.