Cold gases of electronically excited Rydberg atoms and groundstate polar molecules have generated considerable interest in cold matter physics, by introducing for the first time many-body systems with interactions which are both long-range and tunable with external fields. The overall objective of this proposal is (i) the development of theoretical ideas and tools for the understanding and control of non-equilibrium dynamics in these diverse systems and in their mixtures, including dissipative effects leading to cooling, and (ii) to analyse emerging fundamental phenomena in the classical and quantum regimes of strong interactions, leading to innovative simulations and experiments of complex classical and quantum systems. The project is divided into three parts, with strong overlap:1) Rydberg atom dynamics: The study of complex open-system dynamics in gases of laser-driven Rydberg atoms, including the study of the effects and control of dissipation and decoherence from spontaneous emission in strongly interacting gases.2) Cooling of complex molecules in atom-molecule mixtures: The theoretical investigation of novel ways to perform cooling towards quantum degeneracy of generic, comparatively complex molecules, beyond bialkali ones, in mixtures of groundstate molecules and of Rydberg-excited atoms.3) Simulations of strongly interacting many-body systems at the quantum/classical crossover: Atomistic characterization of formation and dynamics of formation of strongly correlated phases with long-range interactions.For each of these subjects, the objectives are at the cutting edge of fundamental atomic and molecular science and technology.