Abstract: We consider two interacting particle systems, Activated Random Walk and Oil and Water, which belong to the so-called class of Abelian networks. In these systems particles of two different types are present and the dynamics conserves the number of particles. This conservation law introduces hard-core constraints and long-range correlations which make the rigorous mathematical analysis particularly challenging. Activated Random Walk undergoes an absorbing-state phase transition, namely if the particle density is small enough, the activity dies out with time, while if the particle density is large enough, the activity is sustained. In recent years significant effort has been made to provide a rigorous proof of the occurrence of such a phase transition and a non-trivial characterisation of the critical particle density. Oil and Water shows a substantially different behaviour from Activated Random Walks and other Abelian networks: the activity dies out with time at all densities for any graph which is vertex-transitive. This talk reviews the main questions, results and conjectures about these models and presents joint works with E. Candellero and A. Stauffer.