Abstract: This talk reviews fundamental principles of contraction theory in dynamical systems, including an analysis of the contractivity properties exhibited by neural networks. Then I will discuss the network contraction theorem and explain why it proves excessively restrictive for several networked systems. Finally, I will present some elements of an emerging theory of semicontracting systems. I will formulate a duality theorem that explains why the Markov-Dobrushin coefficient is the rate of contraction for both averaging and conservation flows in discrete time. If time allows, I will show applications to the Kuramoto model of coupled oscillators.