This proposal embraces the broad theme of mathematical analysis of large interacting systems, which consist of several small components that randomly interact with one another over space and time. This concept arises in many fields, and paradigm examples studied in the probability, statistical physics and computer science literature include percolation, spin systems, and random and dynamic networks. The development of rigorous statistical mechanics and its influence on modern probability theory turned into a remarkable success story in the second half of the last century, not only enriching both fields, but at the same time stimulating and establishing new connections between probability theory, complex analysis, dynamical systems etc. Many powerful theories and techniques were produced on both sides, which led to deep understanding of equilibrium problems, in particular for systems whose local interactions at microscopic level give rise to weak ``macroscopic independence''. In parallel, demands from theoretical computer science, combinatorics and non-equilibrium statistical physics offer a large class of models where local microscopic interactions either produce strong correlations at macroscopic levels, or generate non-equilibrium dynamics, whose behavior changes drastically in time, breaking stationarity and ergodicity. This prevents current methods based on ergodic theory and rigorous statistical mechanics techniques (e.g., energy vs. entropy, finite energy and combinatorial arguments) to be applied, and puts us in front of great challenges. Our overall objective is to develop mathematical techniques to analyze such important and difficult models, producing ground-breaking results in this area, establishing new connections with other topics, and opening up future directions of research. In order to make progress towards this broad goal, we will concentrate on four specific models, which are interesting in their own right, and exhibit important and challenging characteristics and phenomena that are common to a large class of systems.